[Cute question heard elsewhere]

Is there a nice characterization of extensions of fields $K/k$ such that whenever $E/k$ and $E'/k$ are subextensions and $\sigma:E\to E'$ is an isomorphism over $k$, there is a $\tilde\sigma\in\mathrm{Aut}(K/k)$ such that $\tilde\sigma|_E=\sigma$?

Normal extensions and those without proper subextensions have that property. On the other hand, $\mathbb Q(\sqrt[4]{2})$ doesn't.