Potentially normal-subhomomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a potentially normal-subhomomorph-containing subgroup if there exists a group $$K$$ containing $$G$$ such that $$H$$ is a normal-subhomomorph-containing subgroup of $$K$$.

Stronger properties

 * Weaker than::Normal-subhomomorph-containing subgroup
 * Weaker than::Amalgam-normal-subhomomorph-containing subgroup
 * Weaker than::Finite normal subgroup
 * Weaker than::Periodic normal subgroup

Weaker properties

 * Stronger than::Potentially strictly characteristic subgroup
 * Stronger than::Potentially characteristic subgroup
 * Stronger than::Normal subgroup