If $L$ is a Galois extension of $K$ and $M$ is a finite Galois subextension of $L \mid K$, then a standard lemma says that any automorphism of $M \mid K$ can be extended to an automorphism of $L \mid K$; in other words, the restriction homomorphism $\textrm{Gal}(L \mid K) \to \textrm{Gal}(M \mid K)$ is surjective.
Question. Is this true for infinite Galois subextensions as well? If it is, where can I find a proof? If not, what is a counterexample?