How do you befriend a mistreated cat?

You ask a very good question. I think it's a matter of time that your cat will become accustomed to you and learn that you will not hurt her. She needs to trust you, and like in humans, gaining trust takes time. I think if you're really worried, you can take her to the vet and talk to your vet about your concerns. It's possible your vet will refer you to an animal behavioral psychologist who can help you work on your relationship with her. Good luck!

It takes time to befriend an abused cat.  It is not something that will change over night.  You just have to be patient!  When she is affectionate, go with it and pet her.  If she becomes scared and hisses, move away and try to remember exactly what happened or what you did that may have scared her. (in psychology we call this the antecedent event).  And then try not to do it again.  Just lots of love and treats, and lots of time, and she should become a happier cat.  She may never recover 100%, but she should get over a lot of her issues over time.

good luck!

Trust Takes time as the others have said. ur as scared as she is so it will definetly take some time so be patient.  u should keep petting the cat but if she gets up and leaves tht means she doesnt want to b petted.  give her a lot of love and you'll b fine.she will understand tht all you want is to b close and she will start to realize tht ur not scary.  B patient.

Good luck. 

Of course like everyone has said already it will take time.But, I took in a cat like that too, and the best thing i have found is let them come to you, give lots of love when they do, but wait til they choose to come get it.

Time, trust and patience.  That's the secret, as everyone has mentioned.  I took in a young cat that had been abandoned, and we all believe she was mistreated as well.  Same scenario.  One minute loving and cuddling, the next minute her ears were flat back, tail huge and claws out.  Jokingly we nicknamed her 'Face Biter', because you could be holding her and suddenly she would growl and actually snap at your face.  And those claws were wicked.

I left her to her own devices.  Having other cats, she was never alone, but it seemed at night she wanted to sleep in our room.  So we would let her.  She'd stay at the foot of the bed, sometimes venturing up to sniff one of us.  On rare ocassions she's snuggle close and go to sleep.  We'd sometimes pet her, or scratch an ear.  Always talked to her in a soft voice.  This has become a nightly ritual.

Its been almost 9 months and Noelle (her real name) has mellowed out some.  She's more content in her new surroundings.  She comes when she is called.  We can scoop her up and play with her, or scritch along her back when she comes along side us.  When its bedtime, she will make her presence known ... nuzzle us and lick our cheeks or nose, then go take her place at the foot of the bed.

Time, trust and patience.  She will come around.