Amalgam-characteristic 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., amalgam-characteristic subgroup) must also satisfy the second subgroup property (i.e., image-potentially characteristic subgroup)
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Statement

Suppose $H$ is an amalgam-characteristic subgroup of a group $G$. In other words, the amalgamated subgroup $H$ is a characteristic subgroup of the amalgamated free product $G*_{H}G$. 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 $\rho (L)=H$.

Related facts

• Amalgam-characteristic implies potentially characteristic

Proof

Given: A group $G$, a subgroup $H$ of $G$ such that $H$ is characteristic in $G*_{H}G$.

To prove: There exists a group $K$, a surjective homomorphism $\rho :K\to G$, and a subgroup $L$ of $K$ such that $\rho (L)=K$.

Proof: Let $K=G*_{H}G$, and define $\rho :K\to G$ as follows: simply identify the two copies of $G$ and carry out the multiplication. Let $L$ be the amalgamated subgroup $H$ of $K$. Then, by assumption, $L$ is characteristic in $K$ and $\rho (L)=H$.