Potential new members respond to invites but don't get admitted. What gives?
I keep sending invitations to potential new members, they respond to the invitation, but they're not admitted to the group.
This might not be the same thing. One group I joined, I had to join twice. The first time I joined, I got a response telling me that I had joined the email list. Since I wanted to join in order to post paintings and stuff, I had to join by email.
Look at the "Who can join?" preference in the Group Settings. If moderator approval is required then you must approve them using the Pending Activity pages.
Otherwise, have a look in the Membership tab of the group's Activity Log - see if it tells you what has happened with these would-be members.
How 'bout you post a reply?