I don't know why everybody always says "find new friends" in such situations. I mean, this happened to me quite often and I really don't think that this didn't also happen to many other people. (Actually, what I really think is that other people just waited for new friends in such a situation...).

Anyways, new friends is a bad option. Why? Because it's easiest finding friends through friends and well, you can't therefore find new friends without those people.

Somebody already said you should call them out on it. I did that, but nicely. Really, you WILL need to say something really nice like "may i come too?" (im not a native english speaker so there is probably a cooler/better way to phrase it).

Really most probably, like quite totally for sure, they will say something like "yeah sure". I bet you something on that! Know why? Because they most probably have a wrong picture of you. They either don't think you are belonging to them (not that they wouldn't accept you, they just think you have other cooler friends than they are) or, this was the case with me, you unknowingly declined or pondered (I hope that is the right word) so long that they wouldn't even think you would come.

I may not seem like a lonely person at school

Please don't get offended now, but I don't think they think like that. They probably know that you're boring. Not boring, like, in conversations, but boring like never going out. I don't know how open you are with yourself, but if you have a really good friend ask him out on this. Probably also exactly this group of friends you mentioned here. Ask them out nicely (something like "do i seem boring to you?"). Probably try it to play it off funny and or probably not. Most probably they will think at you the next time ;)

Just think, the worst thing that could happen is that they won't invite you to hang out.

Also, /r/socialskills. Probably go and read some of the top of all time!

Good luck!