Double check that it includes feature services. Open the map in arcgis.com and ensure that you can edit the underlying data. If you want this to be taken offline, go into the map item properties and see if you can check the box under enable offline mode.
I also have seen goofy behavior with maps mysteriously being "unshared" with the group. In which case you need to: open the map, unshare, save, then share, then save again. And then it should show up amongst group members.