TITLE: Counter-examples of Galois Correspondence
QUESTION [5 upvotes]: What are some examples of a separable field extension $L/K$ and a subgroup $H$ of $\text{Aut}(L/K)$ such that $\text{Aut}(L/L^H) \neq H$? Here $L^H$ means the fixed field of $H$.

REPLY [2 votes]: For $H$ finite, it is impossible:
Dummit & Foote Section 14.2 Corollary 11 (adapted with your notation):
Let $H$ be a finite subgroup of automorphisms of a field $L$ and $L^{H}$ be the fixed field, then $Aut(L/L^{H})=H$.
In your problem, $H\leq Aut(L/K)\leq Aut(L)$, we don't care about $K$ at all, we still get $Aut(L/L^{H})=H$.