My instant response is no. Also does she want to go? Should I let her implies that she's asked?
I'm going to list below some of the things I put my mum through from the age of 12+, I turned a corner by 15, but I'm surprised she didn't send me into care (she threatened it once and it broke my heart!)
I was out getting drunk sleeping with people at 13. She didn't know this....apart from when I didn't come home and she'd hunt the streets for me.
I smoked from the age of 12.
I got suspended from school 8 times, mainly for bunking off, any chance I could get I'd run away from school.
I shoplifted a lot.
I ran away from home a few times, once she called the police to look for me.
I was generally a little shit!
BUT she never gave up on me. Not once. Please don't give up on your daughter, sending her away will fail her.
I'm just saying this as she isn't a bad kid. I did all that as I was desperate for friends and happened to fall into a bad crowd, so I was desperate to please them. There's so many reasons she could be playing up. Try to find out why.
Also me and my mum went to counselling at the peak of it (took an attempted suicide on my part for it to come to a head, although I only took about 12 paracetamol!) and it reakky helped out relationship. The GP organised it. Can you go down that route??