When I use Administrators group in the Task Scheduler, it works fine when triggered manually (and of course it doesnt trigger automatically because I cannot change the checkbox for "Run whether User is logged in or not").
That's because Administrators (plural) is a group and not a user. A group cannot logon. If task scheduler is running with that group then it is really using whatever account that you are logged on with.
When I run the script from PowerShell As Administrator, it works fine too.
Does that mean that you logged on to the machine with the account named Administrator or does that mean that you selected the "Run as administrator" option to elevate the process for UAC access?
I have a domain user which has Domain Admin permission.
For example, lets call that account MyDomain\MyAdmin. And you have machine AAA where the script runs and you are trying to query the services on machine BBB. Are you able to logon (RDP) to both machines using the MyDomain\MyAdmin and perform administrative tasks? (To verify that both machines see that account as a member of the administrators group.)
Using that account on machine AAA run these commands. Do they work?
get-service -ComputerName BBB
invoke-command -ComputerName BBB -ScriptBlock {Get-service}
Cannot open Service Control Manager on computer 'xxx.xxx.xxx.xxx'.
Are you using the computer name or the IP address on your Powershell commands? I would recommend using the computer name. In fact you may want to try using the fully qualified name, BBB.MyDomain.Com.
Finally, on the task definition you will see a checkbox labeled "Run with the highest privileges". Make sure that is checked. The task should run as your MyDomain\MyAdmin account.