Central implies image-potentially characteristic

Statement
Suppose $$H$$ is a central subgroup of a group $$G$$: $$H$$ is contained in the center of $$G$$. Then, $$H$$ is an image-potentially characteristic subgroup of $$G$$: there exists a surjective homomorphism $$\rho:K \to G$$ and a characteristic subgroup $$L$$ of $$K$$ such that $$\rho(L) = H$$.

Related facts

 * Normal subgroup contained in hypercenter is image-potentially characteristic
 * Finite normal implies image-potentially characteristic
 * Periodic normal implies image-potentially characteristic
 * Central implies amalgam-characteristic

Facts used

 * 1) uses::Central implies amalgam-characteristic
 * 2) uses::Amalgam-characteristic implies image-potentially characteristic

Proof
The proof follows directly from facts (1) and (2).