What do I do to get my new family group member accepted into the group?
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I am just another 'lowly' User/Patron ...
[ And, I happen to ne a Member of the Church ... ]
Plus, I have "Created", a "Family Group"; and, "Invited" and "Approved" a NEW member to that group.
IF, you are the (or, one of the) "Administrator(s)" of the particular "Family Group"; THEN, you must, 'Access' the "Family Group", and, "Approve" the invitee/new member ...
Here are some "Knowledge Articles" in 'FamilySearch':
I hope this helps/assists, somewhat.