Outer Automorphisms of Galois groups Assume $K/\mathbb{Q}$ is Galois extension of number fields and let $F \subset K$. Let $Gal(K/F) \cong G$ and let $\sigma$ be an outer automorphism of $G$ as an abstract group. Is there a way to interpret $\sigma$ in relation to the fields $K,F$?
If $\sigma \in Gal(K/\mathbb{Q})$, then this indeed gives us a way to interpret this action, but what if this is not the case? Of course $\sigma$ won't be a field automorphism but I'm not sure what else I can say.
As an example, consider $f=x^3-7x-7$ over $\mathbb{Q}$, which has Galois group $C_3$. There is an outer automorphism swapping the two $3$ cycles but how does this interact with the splitting field of $f$?
 A: A simple thing to say is that an outer automorphism of $G$ induces an automorphism of the poset of subextensions $F \to E \to K$. For example, if $F = \mathbb{Q}, K = \mathbb{Q}(\sqrt{a}, \sqrt{b})$ where $a, b$ are such that $\text{Gal}(K/\mathbb{Q}) \cong C_2 \times C_2$, then the outer automorphism swapping the two copies of $C_2$ swaps the two quadratic subfields $\mathbb{Q}(\sqrt{a})$ and $\mathbb{Q}(\sqrt{b})$. Unfortunately this isn't interesting in the case of your example because there aren't any nontrivial subextensions. 
There's a more complicated thing to say if you want $\text{Out}(G)$ to itself be precisely the automorphism group of something related to the fields. In that case one thing to say is that it is the group of autoequivalences of the category of finite $G$-sets (up to natural isomorphism), and that the category of finite $G$-sets can be identified with the opposite of the category of finite etale $F$-algebras which split over $K$. Explicitly, the objects of this category are finite-dimensional $F$-algebras of the form
$$A \cong \prod_{i=1}^n E_i$$
where each $E_i$ is a subextension $F \to E_i \to K$, and the morphisms are $F$-algebra maps. It's still hard to see what's going on in your example because the action is trivial on isomorphism classes of objects, though. 
