"SSGs are common, and almost always include a L-class ship that is testing one or several types of unmanned systems. It is not uncommon for CGs to deploy with SSGs "
The inclusion of an L-class ship is a recent phenomenom.
I have not seen a CG deploy as part of an SSG.
See my list:
Stennis Strike Group
Although they have been included in the Neptune Warrior exercises of which there were three last year.
The Pacific Fleet was the first if I remember correctly to add a foreign warship to a CVBG a Canadian frigate.
And yes Atlantic Fleet ESGs and CSGs have been primariily in the Gulf and just transited the MED but have also spent some period in the MED at times but most Pacific Fleet CSGs and ESGs have primarilly transited the Pacific enroute to the Gulf but depending on a variety of variables have operated for longer at times in the Pacific for sure.