If you can see the Edit button when you log in with the administrator account, it means that there is an issue with the security role assigned to the user that doesn’t see the button. By default, the Group Manager security role is applied to the Owner (ManagedBy) well-known security principal and is assigned over All Objects. This means that the permissions granted by this role apply to the user who is specified in the managedBy attribute of each group.
Make sure that the user that you are logged in with is specified in the managedBy attribute of the group you are trying to edit. Should the issue persist, please, post the screenshot of the security role here or send it to email@example.com. Make sure that the screenshot includes the assignments of the role.