The story behind the Tuckers is complex. In a nutshell, the Director, Leonard Church, is their father. The reason why Carol, Taho and Lavernius don't follow his father's last name is to honour the Director's wife, Allison Tucker.
Allison, who had always loved Leonard, left when their eldest child, Carol, was 10 years old. She, being part of the UNSC, had to fight in a major war. The Great War. Against the red aliens. Being a remarkable fighter and supportive wife, Allison left a great impression on Leonard, with Leonard having a vivid memory of her. However, she was killed in action during the war at a tender age of 35.
Her fire team was one of the best teams in the UNSC, consisting of: knife expert Felix, infiltration expert Locus, arch rivals Gretel Kimball and Dolores Doyle, Lawrence Flowers, Victoria Grey and... decisive leader Sarjay Grif. All team members had very good relations with Allison, as she was incredibly friendly. The team was sent to a human-inhabited, heavily forested planet faced with an alien threat. Chorus.
Nobody knew the outcome of the mission, mainly because nobody managed to return. It was reported in a message made by the team that their mission was successful, with the red aliens hiding in one last undiscovered village. The only drawback was... Allison died... killed by the red alien leader, Santa. Her body was supposedly buried next to a jungle temple with them expressing their deep condolences for the family. The supplies in the UNSC were exhausted, stranding the team on Chorus without any transportation. So, the team parted ways and made their own destinies, with Felix marrying Gretel to get Vanessa, who did not know who her father was. Locus married Dolores, with Donald being their child. Sarjay married Victoria, getting Emily, Dexter and Kakaina. As for Lawrence... well... he was obsessed with getting back to civilisation, hoping to get back to his two children: Butch and George.
The death of Allison was not taken well by Leonard, obviously. This caused him to honour her with the start up of the project and by renaming their children. He makes it his best effort to get her back, by means of AI. He also ensures that one of the future missions of the project would be on Chorus.
Back to the story. The Director informs Team A to not kill Lavernius to any extent, despite his annoying behaviour. As per normal, Pilot 478 sends them to a rocky planet near to Chorus called Verse. There they find a dozen unconscious bodies strewn all over the outskirts of the compound where a light blue/teal/aqua-armored soldier stood... waiting... to take revenge.
Lavernius had always been an outcast of the family. Being the youngest, he never got any attention from the others, resulting in his seeking-attention attitude. Actually, Lavernius could have made it to the project if it weren't for the Director's disapproval, fuelling his hatred for the other siblings.
So there he waited... for Team A to make a move... and so they did. Well two of them did. Butch and Carol. Despite being given commands by George, they charged straight in. Carol because she wanted to prove to her father that she should be given the Leadership role instead of George. Butch because he simply wanted to annoy George... and of course to try to get kicked out of the project.
Due to the severe lack of direction and coordination, Carol and Butch were effortlessly kicked aside by Lavernius and his self-made armor. George, seeing that he was two men short, organised a synchronised attack with Taho, whose loyalty came from respect. Carol and Butch were just punching and kicking Lavernius aimlessly, colliding into each other as a result. So, George and Taho made their move, surrounding Lavernius and successfully apprehending him without the use of any weapons whatsoever. They loaded Lavernius onto the pelican that was getting ready to launch where they met Team B.
Team B consists of recruits who are later known as Agents Washington, Wyoming, North Dakota and South Dakota taking the positions of leader, sharpshooter, support and muscle respectively. These four showed a deep anticipation for their life in the project, after seeing the success of Team A after only one week. They were also frightened by the fact that competition would arise between the two teams. Nonetheless, they still made the effort to make friends with them, especially George.
"Hey you with the blue armor" North started the conversation
"Yeah?!" Butch, Carol, Lavernius and George respond simultaneously
"No, I mean Team A leader. Nice to meet you. I'm North and this is my twin sister South. We're the Dakotas"
"No way! Butch and I are twins too. I'm Georgia and he's Florida. Technically they're considered north and south right?" George delightfully replied
"No..."North politely responds,"but never mind, I'm sure we all will get along just fine"
The rest introduce themselves to the other team, forming some sort of awkward friendship that ultimately would dissipate given a competitive situation.