Sorry, maybe I ought to have been more explicit about "By the Galois Connection between G and C.":

https://ncatlab.org/nlab/show/Galois+connection#GaloisTheory

> The Galois theory normally taught in graduate-level algebra courses ... involves a Galois connection between the intermediate fields of a Galois extension and the subgroups of the corresponding Galois group.

Instead of speaking informally, of abuse of Galois Theory, I should have spoken formally, of abstraction of Galois Theory.

Does it make more sense now?

By the Galois Connection between G and C.

Something tells me you are now a sage (in the sense of Philodemus :)

Thanks for the targeted mention of triples and the exercises of varying difficulties, that’s like right o’er the ‘bow, Naval Officer!