Amalgam-characteristic implies image-potentially characteristic

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 $$\rho(L) = H$$.