Endomorph-dominating subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed endomorph-dominating if for any endomorphism $$\varphi$$ of $$G$$, there exists $$g \in G$$ such that $$\varphi(H) \le gHg^{-1}$$.

Stronger properties

 * Weaker than::Order-dominating subgroup
 * Weaker than::Homomorph-dominating subgroup

Weaker properties

 * Automorph-conjugate subgroup, if the subgroup is a co-Hopfian group (i.e., it is not isomorphic to any proper subgroup of itself).