Inner holomorph of a group

Definition

Let $G$ be a group. The inner holomorph of $G$ can be defined as the semidirect product $G⋊\mathrm{I}\mathrm{n}\mathrm{n}(G)$ where $\mathrm{I}\mathrm{n}\mathrm{n}(G)$ is the inner automorphism group with the usual action.

It is a subgroup of the holomorph $G⋊\mathrm{A}\mathrm{u}\mathrm{t}(G)$ and is a quotient of the direct product $G\times G$.

Facts

When $G$ is an abelian group, group of nilpotency class two, group whose center is a direct factor, or centerless group, this is isomorphic to the central product of two copies of $G$ with the center $Z(G)$ of both copies identified: $G{*}_{Z(G)}G$. In other words, it is the quotient of $G\times G$ by the subgroup $\{(g,g^{-1})\mid g\in Z(G)\}$.

If $G$ is a group whose center is a direct factor, this group is isomorphic to the direct product of $G$ and $\mathrm{I}\mathrm{n}\mathrm{n}(G)$.