Not being able to see Group is probably because of the access to the User Administration and Group modules which is usually reserved for users with user_admin, admin, or role-delegator roles. Group in particular, in my PDI, shows requires user_admin role. You could update the application and module navigation items to include the new role so they are visible.

For the access to read/write on the Group records, check the ACLs that grant those rights. You might be able to just add the new role to those ACLs. That would grant access on records without being the Manager though so your use case might require a new ACL for read and write to account for the user having the correct role and the user being in the Manager field.