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 $$\operatorname{Inn}(G)$$.