The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the fundamental theorem of Galois theory (FTGT); $L: H = Gal(K/L), L = K^H$, where $K^H$ is the subfield of $K$ fixed by $H$. Let $\sigma \in G$. Prove that $\sigma(L)$ is the intermediate field which corresponds with the subgroup $\sigma H \sigma^{-1} \leq G$ by the FTGT.
I am expecting normalcy to come up but beyond that intuition, I am stumped.