[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, $\mathbb Q(\sqrt[4]{2})/\mathbb Q$ doesn't.
Dear Mariano, I can't characterize the extensions in your interesting question, but here is a class of examples.
Take for $k$ an algebraically closed field and for $K$ any algebraically closed extension of transcendence degree one.Then if you give yourself a $k$- morphism $\sigma :E \to E'$ it extends to an endomorphism $\tilde \sigma : K \to K$, since $K$ is algebraic over $E$ and algebraically closed. This extended morphism is surjective, hence an automorphism, because $\tilde \sigma (K) \subset K$ is algebraic and $ \tilde \sigma (K)$ is algebraically closed. Since $k$ is fixed under $\tilde \sigma$ you get your result.
I suppose this construction might be generalized somewhat.
