My initial assumption was that people would interact with their groupsThrough email is certainly true for me. Apart from moderator duties, I only go to the group's archives when I'm new to a group or to research messages that I may not have kept in an email folder.
When you subscribe to a group, all of the group messages instantly showInteresting about the Inbox view. I ignored it largely out of inexperience with Groups.io, I see how that's a valuable way of keeping up with one's groups for those who don't rely on email. I would never have thought of it as a way to research the messages of a group I just joined though - I'd want to be looking at just their messages.
I think affordances for Muting or Following a thread likely should appear wherever one sees a thread - in the Inbox list itself (not just after opening the thread/message), as well as in the lists and opened messages of the group's pages.
And there are separate views for threads you've already muted orExcellent. It is possible that group- and member-specific versions of those lists ought to appear as tabs of each group's Archives page. I think that will help bring these features "out in the open" - make their utility more discoverable as well as making them easier to understand. Perhaps there labeled "Your Followed Threads" and etc. to make it clear that those are member-specific.