Hi Mike,
You asked: “[w]hat is the analog of tree-width for finite permutation groups? This should have an answer. And the answer should be fairly obvious/deducible (Cf Grothendieck) from the right abstract point of view.“
This is a companion discussion topic for the original entry at https://bmbumpus.com/2023/11/08/tree-decompositions-of-groups-a-letter-to-mike/