I absolutely hate my mother?

Have you tried to tell her how you feel? Maybe she is shy or afraid to get out of her comfort zone like some have said. I don't get the part about her wanting YOU to apologize to your sis. If my sis did that to me she would be dead to me, your mom needs to stay out of that because it seems she doesn't have the full story. Don't hate your mom, just back off and if she wants to see you maybe she will put forth the effort, if not. don't worry about it, send her a card or call her sometimes.