Let $F/K$ be a finite Galois extension of fields (of characteristic 0 if it makes a difference). Let $G_F$ and $G_K$ denote their absolute Galois groups.
Q: What is the relationship between $G_F$, $G_K$ and $Gal(F/K)$, thought of as abstract groups?
This must be elementary, but I keep getting tangled up in confusion trying to explicitly work out the various homomorphisms and indeterminacies involved. There is a homomorphism from $G_F$ to $G_K$, the image of which is basically everything except $Gal(F/K)$, but I can't seem to get any additional clarity. I suspect that I'm either missing something straight forward or it's hard to say in general. I've been out of the pure maths world for some years so probably the former!
EDIT: my naive guess from finite Galois theory (that $G_F$ is isomorphic to an open subgroup of $G_K$ with the quotient being $Gal(F/K)$) seems to be false due to the example in comments. But I might be confused here too.
The real confusion here, alluded to in comments, was clarified in the following: Galois correspondence for pro-p extensions