I've been having this same problem for well over a year now. I have a 4S and am a Verizon customer. Group messaging works fine as long as all of the other parties have an iPhone. The issue is when at least one of the other parties has a non-iPhone ... In those cases I cannot send or receive group texts. I also cannot send or receive pictures from non-iPhones, even when individually texting. I wish they would come up with a way to fix this!!
This is the solution: https://discussions.apple.com/thread/5323646?start=0&tstart=0
In short, you are out of memory in your SMS Attachments folder and the phone won't accept the additional data from the group message/picture message. There is a program called iBackupBot which will solve the problem. It allows you to delete these attachments from the root of the phone. It isn't free but it is worth every penny. I hope this solves your issue because it solved mine.
This same thing is happening to me. When I have group messages, I receive messages from everyone except any Android users in my group. I've tried everything but it's very annoying. My friend with the same phone as me (iPhone 6) receives the messages fine and I can text the android user 1 v 1 just fine.