Can you grant an Active Directory Group rights on a linked server?
You can do by creating a role for the AD Group user and assigning permissions to it.


Create your Active Directory Group as a login in SQL Server as below :

USE master;
GO
CREATE LOGIN [COMPANYDOMAIN\ADGroup] FROM WINDOWS;
GO

Then create a user from the login in the database

USE DatabaseName;
GO
CREATE USER [COMPANYDOMAIN\ADGroup] FROM LOGIN [COMPANYDOMAIN\ADGroup];
GO

Now Grant appropriate permissions to the user 

Note: Granting a server permission to a database role is not possible 

GRANT EXECUTE ON SYS.XP_PROP_OLEDB_PROVIDER to [COMPANYDOMAIN\ADGroup];