Inner holomorph of a group

Definition

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

It is a subgroup of the holomorph $G \rtimes \operatorname{Aut}(G)$ and is a quotient of the direct product $G \times G$.

Facts

When $G$ is an group having an automorphism whose restriction to the center is the inverse map, this is isomorphic to the central product of two copies of $G$ with the center $Z(G)$ of both copies identified: $G *_{Z(G)} G$.

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