# Galois correspondence and characteristic subgroups

It is well-known that Galois correspondence sends a normal subgroup to a normal extension of a field. Specifically, given a Galois extension $L/K$ and the corresponding Galois group $G$, normal subgroups of $G$ correspond to normal subextensions $F/K$ .

Is there a characterization of the subextensions corresponding to characteristic subgroups?

That question doesn't make sense because automorphisms of $$Gal(L/K)$$ don't have any meaning in Galois theory.