Let $G$ be a given finite group. Is there a way to extend the field $K$ such that for the extension $L\geq M\geq K$ we have that $L/M$ is Galois and it's Galois group $Gal(L/M)$ is isomorphic to $G$ ? If there is, along which lines does the proof run ?
(I didn't manage to come up with a proof - nor with a counterexample to this question)