Ok, I admit that this has a certain amount to do with publicity, but it's an interesting idea. It seems that Power Girl is not entirely herself post the DC reboot. I wondered what might happen to a character who essentially had her personality erased, but who still had great power, and possibly had conflicting partial memories of the past. So I came up with a little story in which Kara / Karen / Power Girl starts out in a hospital, unsure of who she is, but learns more as she wanders around the DC Universe, bumping into her cousin, Atlee, Huntress, Oracle, etc, until eventually she discovers her true nature.
Here's a link: http://www.fanfiction.net/s/7482360/1/Lost_Girl_of_Krypton
Warning: the story is very bleak, and, as I was in a non-canonical universe, I felt free to kill off established characters. Power Girl starts of psychotic and, by the end has passed into something else entirely. And whether any of it is real is an open question.