Conjugate Groups of Galois Group if and only if Isomorphic Extensions

Question

Let $L/F$ be a finite Galois extension. Let $K_1$ and $K_2$ be fields with $F \subseteq K_1 \subseteq L$ and $F \subseteq K_2 \subseteq L$. Let $H_1 = Gal(L/K_1) \leq Gal(L/F)$ and $H_2 = Gal(L/K_2) \leq Gal(L/F)$. Prove that $H_1$ and $H_2$ are conjugate in $Gal(L/F)$ if and only if $K_1$ and $K_2$ are isomorphic extensions of $F$.

Answer 1

One direction: argue $G(L/\sigma K)=\sigma G(L/K)\sigma^{-1}$ for any $\sigma\in G(L/F)$.

Other direction: extend the isomorphism $K_1\cong K_2$ to $L\cong L$.