I invited a person to join a secret group but once they accepted the invitation they didn't immediately see that secret group in their "My Groups" listing. Also in the group Member tool I still saw that there was an outstanding invitation. But the next day (Eastern time) all was good with what the new secret group member saw in their own "My Groups" listing and that I saw the invitation was accepted and they were now a group member.
For another invited member of the same secret group the passing of midnight didn't seem to do the trick.
I wasn't sure if the observed delay in the first case made sense with how the ACS Network manages the group joining and "My Groups" list? In second case we are re-sending the invitation.