I start from the vague question "how should a Galois group behave"?
For example, what can be said for $Ext^1(G,\mathbb Z)$ when $G$ is a Galois group? And what about the behaviour of $Ext^1(G,A)$ for $A$ an arbitrary (abelian?) group?
This is actually still a vague question, perhaps, but I was interested in knowing any relationships between the homological point of view (so $Ext$, but potentially any interesting functor on groups) and the fact of "being or not Galois", and I coundn't find them anywhere. Thank you in advance.