Hi Guillaume deW,
Thank you for posting in Q&A forum.
whoami /groups
Displays the user groups to which the current user belongs.
For more information about the command whoami, please check the link below.
whoami | Microsoft Learn
I have done a test in my lab:
AD group: testgroup1, like MyComputer_AdminGroupdoes in your case.
AD user: t2 (I added t2 to testgroup1and Administrators in domain), like admin_user in your case.
AD user: t2-xiashu
Local user in domain machine (member server): tt1
I added testgroup1 to local Administrators group in domain machine (member server).
ONLY when I sign in using domain account t2 on domain machine (member server) and run whoami /groups, I will see A\testgroup1.
If I sign in using tt1 or t2-xiashu or A\administrator on domain machine (member server) and run whoami /groups, I will not see A\testgroup1.
If you want to check the security list, you can also go to users and computers to check the local admin group members.
You can also check below documents for Kerberos Authentication:
How the Kerberos Version 5 Authentication Protocol Works: Logon and Authentication | Microsoft Learn
Note: In my lab, all the users and domain machines are connected to domain.
Hope the information above is helpful.
If you have any question or concern, please feel free to let us know.
Best Regards,
Daisy Zhou