By a straightforward computation, it is not hard to show that the set $\operatorname{Inn}(G)$ of the inner automorphisms of a group $G$ is a normal subgroup of $\operatorname{Aut}(G)$, see for example this question.
Typically, more insight is gained by identifying a normal divisor as the kernel of some homomorphism. For example, $A_n \triangleleft S_n$ is the kernel of the signum $\operatorname{sgn} : S_n \to (\{\pm 1\},\cdot)$, $\operatorname{SL}(n,K) \triangleleft \operatorname{GL}(n,K)$ is the kernel of the determinant $\det : \operatorname{GL}(n,K) \to K^\times$, and the center $Z(G) \triangleleft G$ is the kernel of the homomorphism $G \to \operatorname{Inn}(G)$, $g\mapsto (x\mapsto gxg^{-1})$.
Now I wonder:
For a given group $G$, is there some "natural" group homomorphism $\operatorname{Aut}(G) \to H$ into a group $H$ whose kernel is $\operatorname{Inn}(G)$?
Of course, a solution is given by the canonical projection $\operatorname{Aut}(G) \to \operatorname{Aut}(G)/\operatorname{Inn}(G)$, $\phi\mapsto \phi\operatorname{Inn}(G)$, but this is not what I'm looking for. By "natural", I mean some homomorphism which is of significant importance on its own, like in the above examples the signum or the determinant.