Amalgam-characteristic subgroup

Definition with symbols
Suppose $$G$$ is a group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is an amalgam-characteristic subgroup of $$G$$ if $$H$$ is characteristic in the group $$L$$ given by:

$$L = G *_H G$$.

In other words, $$L$$ is the amalgam of $$G$$ with itself over $$H$$, and $$H$$ is treated as the subgroup of $$L$$ given by the amalgamated $$H$$ between the two factors.

Stronger properties

 * Weaker than::Finite normal subgroup:
 * Weaker than::Periodic normal subgroup:
 * Weaker than::Central subgroup:
 * Weaker than::Normal subgroup contained in the hypercenter:
 * Weaker than::Amalgam-normal-subhomomorph-containing subgroup
 * Weaker than::Amalgam-strictly characteristic subgroup

Weaker properties

 * Stronger than::Retract-potentially characteristic subgroup:
 * Stronger than::Potentially characteristic subgroup:
 * Stronger than::Image-potentially characteristic subgroup:
 * Stronger than::Normal subgroup:

Incomparable properties

 * Characteristic subgroup: Note that the other non-implication is clear from, for instance, finite normal implies amalgam-characteristic, because not every finite normal subgroup is characteristic.