7 month old and 4 year old don't get along

So my household consists of me, my grandmother, my dad, my little brother, my cat Buddy, my grandmothers cat Miss Kitty, and our dog Nugent. Me and my cousin where up at his dads house one day when we seen two little kittens out in the woods. He had a pile of stuff right on the edge of his yard and the woods and there was 3 kittens curled up inside a small pipe. They were only maybe a week old at the most. Well we gave away 2 of the kittens and I decided to keep one. Well when I brought Buddy home he was good. He gets along well with our dog and they are always playing with each other. My grandmothers cat though wants nothing to do with Buddy and all he does when my grandmother lets Miss Kitty inside the house is chase her around the house. Well today Buddy finally attacked Miss kitty. I'm not sure why he keeps trying to attack her (most of the time i put him downstairs in my room until Miss kitty goes back outside). My grandmother gets scared and freaks out and runs after my cat with a fly swatter thinking thats going to help. What should I do to get them used to each other? My grandmothers cat Miss kitty is ALWAYS outside. She maybe comes in a night once in a while but most of the time its outside, and she is always getting into fights with other cats...she is really almost wild she always attacks me and other people, but won't attack my grandmother. At first my cat would try to go up to Miss kitty and play with her just like he does with the dog, but she doesn't want to play ever. So now he just stalks her every time shes in the house and tries to attack her. What am I to do?

I would post this in the behavior forum as you may get more help there.
Good luck!!!