How to assign rights to access over the network to other computers not domain users?

Question

Active Directory

Domain, enikey as a "user" in the domain. On each client machine, including the local administrator %vasya%. When logging in directly on the client computer with the local admin account %vasya%, the user has local admin rights.
When enikey from his computer without getting up tries to turn in the format \\%client computer%\c$ and enters the user %client computer%\%vasya% he does not have enough rights. How to give Enikey local admin rights to client computers so that there are no such situations, partially lacking rights. I will categorically not include it in the domain admins.

Answer 1

Create another group
Add a group to the local admins of all client computers
Add the necessary enikeys to the group.
But in general it is strange - there should be enough rights. Maybe C$ is not shared?