Amalgam-characteristic implies potentially characteristic

Statement
Any amalgam-characteristic subgroup of a group is a potentially characteristic subgroup.

Amalgam-characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed amalgam-characteristic in $$G$$ if $$H$$ is a characteristic subgroup in the amalgam $$L := G *_H G$$.

Potentially characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed potentially characteristic in $$G$$ if there exists a group $$L$$ with an injective map $$\alpha: G \to L$$ such that the image $$\alpha(H)$$ is a characteristic subgroup of $$K$$.

Converse
The converse is not true. This follows from the fact that characteristic not implies amalgam-characteristic and characteristic implies potentially characteristic. In other words, there are characteristic subgroups that are not amalgam-characteristic. Since any characteristic subgroup is potentially characteristic, we obtain examples of potentially characteristic subgroups that are not amalgam-characteristic.

Proof
Given: A group $$G$$ with a subgroup $$H$$ that is characteristic in the amalgam $$L = G*_H G$$.

To prove: $$H$$ is a potentially characteristic subgroup of $$G$$: there exists a group with an injective map from $$G$$ to that group such that the image of $$H$$ is characteristic in that group.

Proof: We claim that the group is, in fact, $$L$$ itself.

Observe that we can take the injective map $$\alpha:G \to L$$ as the embedding of the first amalgamated factor $$G$$. Under this embedding $$\alpha(H)$$ is the same as the amalgamated $$H$$, which by assumption is characteristic in $$L$$. Thus, we have an injective map from $$G$$ to $$L$$ under which the image of $$H$$ is characteristic in $$L$$.