I never understood what any of Lilly's motives are. She seems to want to survive and be moral but she goes along with most of Lee's ideals and plans(bringing David/Travis in along with Ben despite being adamantly against it, despite saying she doesn't want to steal, she take it anyway?)
Lilly is a very weird character who is very indecisive she goes with it anyways? I guess she realizes that if she can't win she just her opponent feel so they can suffer, too (Evidenced by her giving you the food duty, not shooting Andy, and looking very disappointed with Lee if he choose to take the food) She's just that kind've person. I personally don't like but to each their own.