Left-quotient-transitively central factor

Definition
A subgroup $$H$$ of a group $$G$$ is termed a left-quotient-transitively central factor if the following holds.

For any group $$K$$, normal subgroup $$N$$ with quotient map $$\alpha:G \to G/N$$, and isomorphism $$\varphi:G \to K/N$$, the following is true: if $$N$$ is a central factor of $$G$$, so is $$\alpha^{-1}(\varphi(G))$$.

Weaker properties

 * Stronger than::Central factor