If $K \subset L \subset \overline{L}=\overline{K}$, then is it true that $\sigma\in Aut(\overline{L}\mid K)$ satisfies $\sigma|_L\in Aut(L\mid K)$?

Question

We consider a tower of fields $K \subset L \subset \overline{L}=\overline{K}$. Is it true that, given a $\sigma\in \text{Aut}(\overline{L}\mid K)$, we can restrict it to an automorphism of $L$, that is, $\sigma|_L\in \text{Aut}(L\mid K)$ ? Here $\text{Aut}(L\mid K)$ means the group of automorphisms of $L$ that fix $K$.

This affirmative was implicit in a proof I saw in a paper. I know that both $\overline{L}\mid K$ and $\overline{L}\mid L$ are normal extensions, but I don't know if it helps to see the result (if it is true).