I have a few children in that age bracket, and I can imagine how heart rending your situation is. I’m sure we all have memories of how cruel children can be, and nothing is sadder than a sad child. I didn’t become an outcast until the teen years, and some of that was by choice, but my sister was one from the earliest grades on. Childhood was really tough on her, and if she had lost any of her few friends, who knows what might have happened.
I’m afraid that missing recess and lunch robs her of chances to make friends. If there is any way she could take her Learning Support after school, or perhaps during English class, it might help.
You say she is immature. Is she physically immature, emotionally, or socially? The former is not nearly as serious to making friends as the latter two. Her teacher certainly can’t force anyone to be her friend, and children are even less capable than we adults at the them. However, you could try forming friendships with some slightly younger children, if that is the case.
I also agree with Lyllyan, although karate is just one of many activities. We enrolled one of our daughters in swimming, just to pull her nose out of books, now the whole family is into it. As I see it, outside activities like sports or arts are a godsend. Because they are all participating in the same activity, the kids have something they are all interested in to talk about. At least with boys, doing things together means being friends. Plus, the kids are from school, and so do not have preconceived notions about her. That should help her make friends, and develop her social skills that are undoubtedly rusty. An activity with a wide range of ages might help too. T
he trick is to pick something that interests her, but is different from her current activities. Not all kids snowboard, but all kids agree it is a “cool” activity. It doesn’t even have to be that cool of an activity, as long as there is a significant number of people doing it. (Was anyone else a band geek?) At around that age, one of my sons came home and insisted on playing the violin. I think it was to fit in! :o