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Amy,

1. I don't understand why certain members have to be “approved" to
begin with, when they have been invited via e-mail.

As Frances said, if they didn't accept the invitation and instead requested to join your group, then they will need approval.

Same if they requested to join using a different email address than the one you invited. In Groups.io every subscribed email address has a separate account and is treated as if a different person.

2. Though I am approving the potential member right away, the
“approval” apparently isn’t actually happening. When these members try
to join the forum they are told they are still "pending approval".

This sounds like a confusion of email addresses on their part. That is, they tried to join using a different email address than the one you approved.

3. When I attempt to "approve" them again, it doesn't matter. Nor will
it just let me re-invite them because they are already in the system
to be "approved". This requires me to "reject" the pending member and
start the whole process again.

Rejecting the pending member and starting over is probably piling confusion on confusion.

You need to determine what is actually happening with each individual email address.

The Sent Invitations tab of the Invite page will tell you which addresses you've sent invitations to, and of those which have accepted the invitation.

The Pending Approval view of the Members page will tell you which addresses have requested membership on their own (not replying to an Invitation), but have not yet been approved or rejected.

The Members view of the Members page will tell you which addresses have become members, by either route (invite or request). Pay attention to the badges next to the email addresses, especially NC which flags those who've not yet confirmed their email address. That is distinct from a pending or bouncing member, but has the same effect of preventing them from receiving emails from the group.

You can use the search box on the Activity page to find the history of any given email address. Leave Actions set to All actions, and Date set to Any, then click Search.

Nowhere can I find a list of members who are "approved" or "pending"
next to their name/invite.

Those are two separate lists, both on the Members page. There's a blue button above the list that selects whether you are viewing Members (approved or accepted an invitation), Pending Approvals, Bouncing, or Banned.

Don't be confused by the Pending page - that is a list of messages that await moderator approval. It has nothing to do with whether the sending address' membership is pending or approved.

Shal
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