Google "the introverted child"
I think they key issue is is she happy? Does she like socialising at school, then coming home and having some alone time? Many people do, and if that's what she wants to do then back off.
I was like that as a teen. Socialising at school was more than enough, and I just found it exhausting the thought of having to deal with more people after school too.
My mum thought the same as you- No friends, should be out doing stuff, meeting people. She was constantly on my back, why don't you ring x, ask y to to this, z is nice, why don't you ask her round. She meant well, but honestly all it did was make me feel like there was something wrong with me, and I always felt a little voice saying that I was abnormal whenever I was alone and happy. I constantly felt I should be "making an effort".
Leave her be. I have an introvert daughter and what is helping her is telling her there's absolutely nothing wrong with being alone if she wants, saying no etc. She seems to be happy knowing her social life is on her terms, and nobody thinks she's a sad loser if she wants to stay in.
With regard to extra-curricular- team activities might be logical to an extrovert to force her to make friends, but I'd look for more individual stuff. Then she can go along, get on with it, and if she's not immediately involved in the social side it won't stand out. Stuff I enjoyed was First aid, (the red cross had a teen night once a week- that's good because you are told how to interact with people, and a lot is role play) swimming club, dance, drama, teen gymnastics, anything where I didn't have to "organise" or fit into a team.
But please, please, don't make her feel like she should have a big social life. She might not want one, and that's OK, really it is. I still don't have a social life, and I love it like that.