I feel this way and have been told i am by several people.
I wish i were invited to more places and considered an essential member of a group.
Hi there Nomad.
It doesn't matter what you have been told by other people. They not understand that by telling you so, they are putting you down, or they be have malevolent motives of prefering you to be isolated or a zillion things...
You say you wish you were invited and becoming a member of a group. You don't have to be invited. No, I am not saying that you have to show up in parties or elsewhere uninvited

(besides someone might call the police if you do that). I am saying that you have to create your own luck, i.e. create opportunities and then take advantage of them. For instance enroll in a language cource to learn a new language. That's a place to meet people, socialise and arrange to go out and have fun. Think of other possibilities too, like classes for Tae Kwoe do, role playing games Clubs, excursions etc
I know what it feels like. I would probably give almost anything to belong to a group, be invited out more often, and have friends. I'm a homebody but that doesn't mean I wouldn't mind going out every know and then.
The same to you transendedphoenix (Jennifer), start making your own luck. Don't expect someelse to save you. Make your own luck. I am sure you can. Go for it!