Automorphism group of the Galois Group

Question

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) as it pertains to $F$ and $K$? I realize this is quite broad, but I suppose that I am curious as to whether the automorphisms of the Galois group can be interpreted in a natural way in the context of the underlying field(s) and vice versa.