On the profile tab, is there a way to have a custom field that auto populates from the group admin, shows as read only for group members, but is still editable by the group admin? An example would be a company logo.
You can edit a custom field, set Populate Data From Group Admin parameter to Yes
If you do that, when group admin add a new member, data for that field will be populated from group admin data. However, group members can still edit and update data of that field on his own user profile page