Direct factor over central subgroup

Definition with symbols
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is a direct factor over central subgroup of $$G$$ if it satisfies the following equivalent conditions:


 * 1) There exists a subgroup $$A$$ of $$H$$ such that $$A$$ is a defining ingredient::central subgroup of $$G$$ and $$H/A$$ is a defining ingredient::direct factor of the quotient group $$G/A$$.
 * 2) In the quotient map $$\rho:G \to G/Z(G)$$, where $$Z(G)$$ is the center of $$G$$, $$\rho(H)$$ is a direct factor of $$\rho(G) = G/Z(G)$$.