Mum won't move on? How can I help her?

My mum has a daughter and son from a different father than me. They have always been distant from us and refuse to have anything to do with us. This upsets mum. (not me since we never got along and they were hardly around). Mum has tried to build a relationship with them. Sending emails..etc.They only send emails when they send photos of their children, which mum has never met. Just to piss her off and upset her more. They know how sensitive my mum is. They moved out of their home when they were very young and deserted mum. Mum isn't a bad person. She is very caring and would love to see her grandkids. Mum always looks them up online and complains about my brother's wife he keeps my mum away from her son. But I try to tell her, if he really wanted to see her, he would. His wife wouldn't hold him back. My sister has also had nothing to do with us for years. But even when she did, she ignored my mum and paid attention to me. They never bought mum gifts, only sometimes. I've tried telling mum that it's their problem, not hers. But she stills tries to email them, but they address my mum like she is a peice of crap. Like, "HI MOTHER" and say some things about their lives and sign-off. Mum now has done something sneaky out of spite to 'test' them. WITHOUT telling me. I was wondering what she was doing, so I came up the screen (we are very open with each other). But she told me to go away and and didn't want me to know. But she told my dad. I was upset. I'm nearly 19 and I understand how she is feeling. She said to me, "I know what you'll say". I am upset she done this, because everything sneaky she has done has backfired and ended with her crying me in tears. I said, "Don't come crying to me! Think before you do it!". She smartly replied with, "YEP!!! I won't come crying. I thought about it". But I know that what she done won't work. 'testing' someone won't get you anywhere. I found the email and i'm not happy. This is just going to backfire. What can I do? How can I get her to move on that doing this won't get rid of the pain? I don't won't her to end up in tears again. My stepdad isn't a sensitive person. She has only been with him for two years, so he hasn't seen what my half brother and sister have treated my mum. He only relies on what she tells him.