Central implies image-potentially characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., central subgroup) must also satisfy the second subgroup property (i.e., image-potentially characteristic subgroup)
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This fact is related to: NIPC conjecture
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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. Central implies amalgam-characteristic
2. Amalgam-characteristic implies image-potentially characteristic

Proof

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