The power of her love to save Harry was huge. She was chattel to James and Serius. Harry had been Serius's, so James Blood Adopted him. Her soul split when she was hit with the killing curse. This is actually what creates ghosts. The Soul splitting in death. Usually from something undone that was so important they split into a seventy thirty or less ghost. Lily was a bit different due to the Chattel Bond and Runes in the house.
The large part of almost eighty percent, went to 'Heaven'. The other less than twenty two percent went to the only being she loved with all her magic, power, and soul. So when her runes activated, but another magic reflected the killing curse. 'She' was right there in front of him. Wishing he would be powerful to disrupt this Wizarding World and love all the girls he wanted.
The 0.78125% of a whole soul flew right into her and both went into the forehead rune she had done on her Harry. She absorbed, the to her, very small fragment. 'She was over twenty five times more powerful as a free standing soul. (The difference was 1.25 times greater, So .25 X 20 = 4.) What she learned was not all sugar and sweet. Much was dark, but then part of her was dark from her enslavement that the teachers did not warn her about. Of course most wives and daughters are slaves to their families. The Bones, Longbottoms, Prewitts, and Zambina's were the exceptions.
It took her a few years to get rid of the tortures and things like that out of her growing soul. She made sure she did not take Harry over. But she sent him sweet dreams. It was at age six, when her step sister hit him in the head with the frying pam. Giving him a major concussion. That 'She' took control of her love. The accidental magic she used blew the gas lines, teleporting Harry to St. Judes in America with his now minor concussion. Over thirty beings felt the power of the explosion. Only Zatara and her dad, the Flamels, Albus Dumbledore, and the ICW Unspeakable Head Researcher (Xeno Odd Lovegood) understood, at the time, it was magic.
Albus had been in Switzerland and so his devices destroyed his office in a fiendfyre type burning. Fawkes had died, because he was still a chick after a burning day, the blood magic, his enslavement bond (which stopped him from living), and the many dark things in the Headmaster's suite, freed him. The enslavement Bond was to keep him Albus's or for him to cease to exist.
Albus had thought Tom needed to kill Harry, so the Prophecy was wrong. Or maybe Neville would work. Odd, his wife Pandora Selene and his daughter Luna Lillyanne moved to the West coast of MACUSA. The American territories had no problem as they were ecstatic to get a researcher, such as Pandora. Xeno supposedly went looking for a Sasquatch tribe in the Redwoods.