Let $N$ be a subgroup of a group $G$, $H := \operatorname{N}\left( N \right)$ the normalizer of $N$. There is a natural morphism from $H$ to $\operatorname{Aut}\left( N \right)$ given by $h \mapsto \hat{h}$ where $\hat{h}\left( n \right)= h^{-1}nh$.
Is there a name in the literature for the image of this morphism or, equivalently, for the quotient $\frac{H}{\operatorname{C}_{H}\left( N \right)}$? Or for the elements in the image?
The specific context I'm interested in is that I'm studying the automorphism group, $\operatorname{Aut}$, of some structure, which has a normal subgroup $N$. Since I have information on the quotient $\frac{\operatorname{Aut}}{\operatorname{C}_{\operatorname{Aut}}\left( N \right)}$ I need to give a name to it. Any suggestion from analogous situations?