I think this is covered in the settings under your group at manage/interaction:
Posting
Who can @mention this group?
Who can message this group?
The workflow you described is intentional, and likely used more widely; we use it as a group inbox/support channel, which requires staged users (non-forum folks, like customers) to be able to email in to a private group used by internal support people.
If I were you I’d look at the forwarded emails behaviour setting. You maybe want “quote” or “hide”, depending on what your group is for. 