Check if the permissions table for user roles shows the appropriate permissions for Group Cover, and that they are activated. I note that this plugin when installed, its use for all remains prohibited.
Also check the file yoursite.com/ow_plugins/groups/plugin.xml for version 10.800. With this version - which is the latest - it works well, with other versions I have not tried it.
If it's not this, I can't think of anything else to recommend you investigate.
It would be necessary to enter the database, check that the records associated with the installation of these two plugins are correct by comparing them with another website where it works, see the log files, check that gheader has created the gheader_cover table correctly, and if No, well, start to insert the step marks in the source code and see variables, where the execution is going and ... It is already more complex but everything has an arrangement.