Inner-centralizing automorphism

Definition
An automorphism $$\sigma$$ of a group $$G$$ is termed an inner-centralizing automorphism if it satisfies the following equivalent conditions:


 * 1) It is in the centralizer (inside the automorphism group of $$G$$) of the inner automorphism group of $$G$$, i.e., it commutes with every defining ingredient::inner automorphism of $$G$$.
 * 2) It preserves every coset of the center of $$G$$, and hence induces the identity map on the quotient group of $$G$$ by its center. In other words, it is in the kernel of the natural map $$\operatorname{Aut}(G) \to \operatorname{Aut}(G/Z(G))$$.

The inner-centralizing automorphisms of a group arise from homomorphisms from the abelianization of $$G$$ to the center of $$G$$. If $$\alpha$$ is such a homomorphism, we can attempt to define a homomorphism as follows:

$$g \mapsto \alpha(\overline{g}) g$$

where $$\overline{g}$$ is the coset of the derived subgroup of $$G$$ containing $$g$$. Such a map is always an endomorphism; however, it need not be an automorphism. If there is an element in the kernel, it must be in the center of $$G$$. If we denote by $$\theta$$ the map induced on the center by $$g \mapsto \alpha(\overline{g})$$, we obtain that the map is an automorphism iff $$g \mapsto g\theta(g)$$ is an automorphism of the center.