Finite normal 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., finite normal subgroup) must also satisfy the second subgroup property (i.e., image-potentially characteristic subgroup)
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Statement

Suppose $H$ is a finite normal subgroup of a group $G$. Then, $H$ is an image-potentially characteristic subgroup of $G$: there exists a group $K$ with a surjective homomorphism $\rho :K\to G$ and a characteristic subgroup $L$ of $K$ such that $\rho (L)=H$.

Facts used

1. Finite normal implies amalgam-characteristic
2. Amalgam-characteristic implies image-potentially characteristic

Proof

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