I created a Global Action that creates a record for a custom object. It shows up on my mobile correctly. It shows up correctly in chatter on SFDC. The custom objects in question both have Read and Create permissions for the users this issue is affecting. These users can use the chatter action on SFDC but it does NOT show up on the mobile app (latest version = 4.02). It only shows up for System Admins. If there is a specific setting preventing thi from whoing up, please let me know!
