It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of endomorphisms something of the sort, and consider an automorphism of $Y$: then it comes from an automorphism of $X$, i.e., it is inner.
Here are some examples which come to my mind:
$\operatorname{Aut}(\mathfrak{S}_n) = \mathfrak{S}_n$ for $n\neq 6$ (here $X$ is a finite set with $n$ elements and $Y$ is $\mathfrak{S}_n$).
The Skolem-Noether theorem that automorphisms of the ring of linear endomorphisms of $k^n$ (with $k$ a field) are inner.
The Dyer-Formanek theorem about the automorphism group of the automorphism group of the free group with rank $n$ (which I was “reminded” of by this question).
We might even argue that the Yoneda lemma has a similar flavor (as it tells us that a homomorphism of homomorphism functors comes from a homomorphism of the underlying category).
I'm sure there are many more, so I think it might be interesting to make a big-list out of them.