Some people like me, but their social GROUP doesn't

Some people like me, but their social GROUP doesn't Started a new job a few months ago. Tried to join a few circles, but I stutter and am awkward because of that (at least it feels like it plays a strong role) I've been rejected and **every time they see me they look away or turn the other way and face towards each other and pretend they don't see me.** This has also happened with lots of individuals that I met one on one, and they ignore me when I see them on the lunch floor. I have made a few friends one on one though. However, when I want to hang out with them at lunch, very often they politely say they can't because they want to hang out with their big group of friends, not in a rude way and I understand that they'd rather hang out with a big group than just 1 person, and so I can't seem to hang out with them all that much. **It feels like I'm fighting for my friends' time with their larger groups of friends, and I just can't win.** They invited me out with the group earlier and tried their best to integrate me but it was obvious the group didn't like me. I also feel like a social burden, and it feels like other people are doing me a favour by being seen with me/hanging out with me. People also don't really sit next to me at lunch, and I usually just have it by myself. It really sucks.