This story is a continuation of my story Superman: Last Son of Krypton. However it's roughly equivalent to the first two movies so this can work on its own.
Like that one, it's set in the 80's. This one, however, does not follow the Supergirl movie, which was awful. So I have a bit more freedom to follow the comics and other media, to find what I feels works best as a quintessential Supergirl story. I hope you like it.

Kara Zor-El sighed as she entered her lab. She was still getting used to it, after years of education finally getting a place to work on her own terms. She loved it to bits. She got out an energy bar and started eating as she got started. She was working on a new system of nanotechnology. Tiny machines that could create and modify battle armor on the fly. Not that Kryptonians did much fighting, but still, this would allow people to put on armor in seconds. Kara looked down at her suit. A traditional white scientific piece with the red and yellow symbol for the house of El. She would love to try on a piece of armor. She'd always admired the rugged yet elegant look of it. She wondered how it would go with long blonde hair. She'd get her chance to see when it came to test out her invention.

It was only half an hour before alarms suddenly blared. Kara looked around, alarmed, and saw the cause- out of the window, she saw some kind of gigantic skull, floating in the clouds above Argo City, with what looked like metal tentacles sprouting from it. Kara didn't know what to do. In emergencies, scientists were supposed to find safe haven.
Kara's father, Zor-El, ran in the room, looking out of breath. "Kara! Come, quickly!"
Kara ran to him and asked, "What is that ship, father?"
He led the way, walking briskly. "It's an entity called Brainiac. It's a... collector, of sorts. We've been aware of it for some time. We need to leave. We have escape pods ready."
"You were expecting it? You knew we'd be attacked?"
Zor-El just said, "Not exactly. It doesn't matter now."
They reached a chamber with a single pod. Zor-El explained, "There are others. We'll launch you, then get in our own pods."
Kara hurriedly got inside. The pod was large, with crystals sprouting from the main hull. The hatch started to close as Alura, Kara's mother, entered the chamber.
When the hatch was closed, it went transparent and Zor-El told Kara, "I'm sorry, Kara. I lied. There are no more pods. I swore that we would not leave, as did Jor-El. You will be alone, except for your cousin Kal."
Kara cried out, "No! Father, stop! I won't leave you!"
He shook his head sadly. "There's no choice. Either you leave, or you face the same fate as the rest of us."
Alura and Zor-El both put their hands on the hatch. Alura said to her, "We love you so much, Kara. The world you go to will be richer to have you in it."
"No..."
Kara's protest faded as the pod launched. She saw the craft rise above the landscape of Argo City, and leave behind the horrible skull that threatened them. Soon, the hatch went opaque again, and she found herself drifting off to sleep as the stasis took her.
Her last thought was for her cousin, who would be relying on her to take care of him wherever they ended up.

Clark Kent landed at the White House and was immediately escorted inside. He was, of course, on time, and the President wanted to see him immediately.
Clark walked into the Oval Office, filled with awe. That office held so much history. The men who had sat behind the resolute desk had molded the country more than any others. The current inhabitant rose and walked around to greet Clark. He extended a hand and shook the President's hand. "It's an honor to meet you, sir."
President Mason looked surprised, but replied, "The honor is entirely mine, Superman. Your reaction to this office surprises me, though. The other Kryptonians were... less respectful."
"I consider myself an American, sir."
"I'm glad to hear that. Please, sit." Mason gestured and sat down behind the desk.
Clark did so, and Mason said, "I have something for you. This might be something of a revelation for you. And..."
He looked uneasy for a moment, then continued, "I would like to ask you to keep it between us. As a patriot."
Clark raised an eyebrow but didn't say anything. He didn't know what to say to that, but the President continued without waiting for an answer anyway.
"In '62 we got a report of something landing in Siberia. An operative in the Kremlin intercepted the information and gave us the location. What we recovered was a craft. Not from this world. Recent discoveries allowed us to tell what was inside, but we've been unable to physically open it without potentially damaging the contents."
He paused for a moment. "Best we can tell, it's a young woman. Maybe dead, maybe in some kind of stasis, we're not sure."
Clark took in this information. It was a lot to absorb. Someone else made it off Krypton! Around the same time as Clark, as well.
Mason gave him a moment, and then said, "I thought you might be able to help us get it open."
Clark looked back at him and replied, "If it is Kryptonian, I should be able to help, yes."

Clark landed outside Area 52 with some trepidation. He wondered what else they were hiding there. Were there other alien craft? He considered x-raying the place, but thought better of it. He didn't need to know. And anyway, no doubt some of the facility was underground and lead-lined.

It wasn't long before an escort came. A General came out and introduced himself as General Franklin. They gave him a ride to the building with the pod in it. It was in a hangar, as it was too big to fit anywhere else.
Clark knew it was Kryptonian as soon as he laid eyes on it. Almost identical to his own, except larger. There were several scientists working around it, and one came over to introduce himself. "Superman, hello. I'm Dr. Wright, I'm in charge of this project. Although not much was going on with it until recently. You could say there's been renewed interest. We're interested to see if you can help us out."
Clark nodded. "It's definitely Kryptonian. I might be able to open it."
He went to where there was a control panel. He pressed the control, but nothing happened. "Yeah, we thought of using Kryptonian DNA to open it, from the others. But no dice. We suspect it's been damaged. Maybe by the crash."
Clark x-rayed the ship. As the President had said, a young woman. She looked to be in good shape physically. "She's alive, but in stasis. If we try to open the craft physically, it could kill her."
The General had been silent thus far, but Clark went back to him and told him, "I'm afraid there's nothing I can do here. I need to take the pod elsewhere."
Franklin frowned. "That's a tall order. I'll need to clear it with the President."
He went off to make the call, leaving Clark staring at the girl in the pod. Who was she, he wondered? She was beautiful, like all the other Kryptonians he'd seen. She reminded him of the pictures Jor-El had shown him of his cousin, Kara.

The thought suddenly struck him. She didn't just look like Kara, She looked exactly like her. But how? Kara disappeared along with Argo City, attacked by a strange skull ship that took it and Kandor weeks before Krypton was destroyed.

Clark was still thinking about it when General Franklin said his name. "The President said yes, under the condition that if you can't open it, you give it back to us."
Clark nodded and thanked the General, his mind still buzzing.

The Fortress of Solitude was silent, as always. For someone like Clark, that was an achievement. Total silence was something rarely experienced except in space, but that was a false silence. This was the silence of nothing happening. It always gave him a sense of timelessness, like the world had just stopped.
But everything was ready. Jor-El had repaired the pod, so all that remained was to open it.
So why did he hesitate? To have living, breathing family from Krypton was a dream come true. And yet, he was afraid. Because of Zod and the other soldiers, or because he'd had restraint drummed into him from such an early age. Would Kara be able to interact with the world without destroying it? Would a handshake end up pulverising someone's arm? A sneeze, end with lethal consequences?

Clark took a deep breath. He'd have to trust her. And teach her. But he couldn't leave her in there. Clark told Jor-El to start the process, and after a couple of minutes, he indicated that it was finished.

The hatch opened, and Kara got out. She was a bit unsteady, and Clark had to prop her up to stop her from collapsing. After a moment, she looked at him, confused. "I don't... who are you?"
"I'm your cousin, Kal. You were in the pod for longer than you expected. It was damaged in the crash."
Kara pushed him away. "No. Kal's a baby. Why are you wearing that?"
Kal moved forward again and Kara pushed him again, more forcefully. Kal flew backwards and into the wall. Kara looked at her hands. "How?"
She looked around, confused, and then turned and ran out of the Fortress.

Kara looked around and saw nothing. Just desolation. Was this the world she'd been sent to? An ice planet?
But no, she could breathe, and the sky was blue. It had to have plants to synthesize oxygen.
She didn't understand any of it. That man wasn't Kal. She needed to find Kal, he needed her. That baby was the only family she had left.
The thought struck her. They were gone. Her family, gone. She fell to her knees and tears ran down her face. She heard a sound behind her. Footsteps, and... his heartbeat?
He said to her, "I'm sorry, Kara. I really am. Krypton is gone. It was destroyed."
Kara sat on the ice. "By Brainiac."
The man seemed uncertain. "No, actually. I don't know what that is. All we know is a giant skull ship took Argo City and Kandor. A couple of weeks later, the planet was destroyed by... something else. My father theorized it would happen, so he and Zor-El built pods to evacuate. I had no idea you'd managed to get out."
Kara sat there for a moment, staring at the frozen wastes. "Are you really my cousin? Was I really stuck in that pod for that long?"
"I'm sorry. Yes. It was damaged and no-one knew how to open it until I revealed myself to the world."
He sat down beside her. "I don't know how much Zor-El told you, but... the yellow sun, and the atmosphere here... our cells can absorb the energy. Use it. We can do things we never could on Krypton. I think that's why our parents sent us here."
Kara couldn't absorb his words. She just felt numb. Everything and everyone was gone. Even that baby was gone.

She sat there for a while, and Kal sat with her, silent. She couldn't tell if it were minutes or hours later that she finally said, "I should be cold."
"We don't get cold. Or hot."
"It feels wrong."
Kal didn't respond to that. For a while, Kara didn't say anything else. But then Kal said, "My father created a shadow of himself. Do you want to speak to him?"
Kara thought about it for a moment, then nodded.