C.B.
She continues to do it, because she's getting the response she wants - upsetting the entire house.
I agree that she needs to experience this herself and learn some empathy. If I were you, I would tease her a bit about something she really cares about and then talk to her and try to get her to understand that this is exactly how you make other people feel. If she still persists in doing it, then the very first time she teased another child she would be sent to her room for the remainder of the day and night - she comes out again tomorrow to go to school. Tell her she does not have the RIGHT to make people feel that way and if she wants to persist, she will have to stay away from people. Plain and simple. You DO NOT have a right to do this to anyone!
She doesn't do it at school because she apparently knows it's wrong and she doesn't want to get into trouble. She KNOWS what she's doing.
Sounds like you have a bully in training. Need to get her under control NOW!