SCOTUS decides what cases they’re going to take (though who knows what arm-twisting goes on behind the scenes). The administration usually does NOT argue in cases because they’re usually not one of the parties. They often are, however, and those they do show up to argue, such as whatever the Osamacare mandate case was called.
Interested, but not directly involved parties may file amicus (”friend of the court”) briefs arguing for one or other finding, and administrations sometimes do that as well.
>>SCOTUS decides what cases they’re going to take (though who knows what arm-twisting goes on behind the scenes).<<
This is called “granting certiorari.” The SCOTUS can choose NOT to do so silently or state why as they deem fit.