Can someone give me advice on this situation?

Im trying to get to know one of my new friends so I want to take her shopping. I don't want to sound mean or anything like that because Im a very friendly person. The thing is, she's overweight and she doesn't have close friends to hang out with on her free time and I think it's because of her weight that people don't want to hang out with her. I know this because I have other friends: I'll be eating lunch with Sophia (the girl im talking about) and ill see my other friends walk up and say hi and then theyll leave and ill see their reflection from the windows cracking up and theyre still pretty close from us so we can hear them say, "oh my god i can't believe shes sitting with her". Theyve never met Sophia before so obviously they said that because shes overweight not because of her personality since they don't know her. So my point is is that I want to become close friends with her, I just care too much about what other people will think of me hanging around a "fat" girl as what other people would say. Any advice?