Let $E/K$ be a finite Galois field extension. Then by the fundamental theorem of Galois theory, there is canonical bijection between the subgroups of $\mathrm{Gal}(E/K)$ and the intermediate field extensions $E/L/K$, sending a subgroup to the intermediate field, whose elements are fixed by the automorphisms of the subgroup.
Question: What is known about the converse statement, i.e. if for a finite field extension, the described correspondence holds, is it Galois?