I was reading the answers to a post, and there was this result that was stated (under certain assumptions that I can't present here) : "If $G$ is the Galois group of $L/k$, and $H$ is a normal subgroup of $G$, then $G/H$ is the Galois group of $K/k$ where $K$ is the fixed field of $H$"
What I managed to do was show that $G/H$ will be embedded in $Gal_k(K)$ (I don't know if that's the right notation, I don't know much about Galois theory), but I could not show that they were isomorphic.
So my question is : under what conditions is the natural mapping $G/H\to Gal_k(K)$ an isomorphism, and how to prove it's surjective in those conditions ?