Let $K\subseteq F$ be a field extension and let $G = \text{Aut}_KF$ be the Galois group of the extension $K\subseteq F$. Let $\mathcal{F}$ be the set of all intermediate fields $L$ and let $\mathcal{G}$ be the set of all subgroups $H$ of $G$.
Define the maps $\Phi: \mathcal{F}\to \mathcal{G}$ and $\Psi:\mathcal{G}\to \mathcal{F}$ by $$\Phi(L) = \text{Aut}_LF = \{\sigma\in G : \sigma(a) = a~~\text{ for all}~~ a\in L\}$$ and $$\Psi(H) =\{x\in F: \sigma(x) = x~~\text{ for all}~~ \sigma\in H\}$$
I can see that the maps $\Phi$ and $\Psi$ satisfy the following properties but I have failed to write proper proofs:
1) $H\leq\Phi(\Psi(H))$, and $M\leq \Psi(\Phi(M)), ~\forall H\in \mathcal{G}$ and $\forall M\in \mathcal{F}$
2)$\Phi(M) =\Phi(\Psi(\Phi(M))),~\text{and}~ \Psi(H) = \Psi(\Phi(\Psi(H))),~\forall H\in \mathcal{G}$ and $\forall M\in \mathcal{F}$