So, I am not certain if this is the correct behavior for the group submission app. Group users cannot submit any listings from the story form only the owner of the group. Not even a group admin can submit a listing.
I tested this with one group "Test Gruppe 1" in the category "Allgemein". I have three profile types and for this test I used the profile type "Gratis Konto". All should be set up correctly I think however the test user cannot submit any listings. I also made a short video.