You need to add the basemap to the basemap group within the organization. Log in as your admin. Create a new map.
Add a layer from the web and make sure the Use as basemap is checked.
![](/legacyfs/online/214063_pastedImage_0.png)
Save the map and share in the Basemap group.
Then it should show up in the basemap group.
![](/legacyfs/online/214133_pastedImage_1.png)
From there it should be in the basemap gallery.
![](/legacyfs/online/214134_pastedImage_2.png)
Then users can select an esri basemap and come back to your custom one.