Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every Galois element gives an automorphism of $K^\times$.
Question: Under which conditions can it be extended to an automorphism of $GL_n(F)$? How?
Cyclic extension, abelian extension, solvalabe extension, general extension?
I am mostly interested in the case, where $F$ is a local field.
Example $\mathbb{R} \subset \mathbb{C}$: We fix an $\mathbb{R}$-basis $\\{ 1,i \\}$. The multipliaction by $a+ib$ correspond to the matrix $$ \begin{pmatrix} a & -b \newline b & a \end{pmatrix}.$$ The Galois element is complex conjugation and corresponds to transpose on the above matrices. Can it be extended to the group $GL_2(\mathbb{C})$?
Motivation: I am actually hoping for an explanation of the Caley transform introduced here on page 2: http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=162025&vfpref=html&r=28&mx-pid=237707
Maybe I am wrong, but consider rather the more canonical group $\mathrm{Aut}_F(K)$ of $F$-linear automorphisms on $K$. The Galois group naturally embedds and acts as inner automorphisms. The elements of $K^\times$ correspond to homothecies and the (inner) Galos action on the homothecies correspond to the natural action on $K$.