I was wondering, when the restriction map $Gal(L/F) \rightarrow Gal(K/F)$ is a surjection? I found a good answer to this question here. In point 3), Starfall explains why with the hypothesis of the theorem, the automorphism extends, and I understand that in order to extend the automorphism we need that $L/F$ is Galois. But why we need $K/F$ to be Galois? I also posted a comment there but maybe I won't get a response.