Suppose I have fields $F \leq L_1 < L_2 \leq E$, where $E/F$ is separable, $[E : F] = p^n$ for a prime number $p$, and $L_2 / L_1$ is Galois and of degree $p$. In addition, I have a subfield $F \leq K \leq E$. Is it true then that $L_2 \cap K / L_1 \cap K$ is Galois (and of degree $1$ or $p$)?
I have proven this in the analogous case for groups, where you replace Galois-ness with normality. However, that proof relies on the Second Isomorphism Theorem, and I cannot find a way of adapting it to the case of fields. As a partial result, $E/F$ is separable, so $L_2 \cap K / L_1 \cap K$ is separable as well, so we only need to show it is also normal to get Galois.
Edit: This is my attempt at solving a more general question I asked here, so it may turn out that this statement is false if this isn't the correct approach.