It would be much more convenient to have ready access to your groups via a drop down menu, the way you do for "Senders", only have another one for "Groups".
That's a reasonable thing to add. Please write up a Feature Request.
I believe the obscurity of this feature may have contributed to problem of IG conversations taking place on the forum.
I believe this never happened, since I have yet to see anyone give a skerrick of evidence of it.