Almost everybody is waiting to be invited, which is how so many people end up feeling left out of a group where nobody actually excluded them. The person who suggests things is not better liked, they are just the one willing to risk a no. That risk is the entire cost of it. It feels much bigger from the inside than it looks from outside.

Send one message this week proposing something specific with a day attached. Vague plans die and specific ones survive. Do it three or four times before you conclude anything about whether people want you around.