Let $F < E < K$ be a tower of field extensions and assume $K$ is a normal extension over $F$. It's well-known that any embedding in $E$ fixing $F$ can be extended to an isomorphism in $K$. One question raised in my mind is that "Can all automorphisms of $F$ be extended to an automorphism of $K$?". The answer is obviously no, for example, the extension $\mathbb{Q}(\sqrt[4]{2})$ is normal over $\mathbb{Q}(\sqrt{2})$, but there isomorphism sending $\sqrt{2} \to -\sqrt{2}$ can not be extended. So my question is under what condition we have all automorphisms of $F$ can be extended to an automorphism of its normal extension?
Any helps would be appreciated! Thanks