Normal-subhomomorph-containing subgroup

Definition
A normal subgroup $$N$$ of a group $$G$$ is termed a normal-subhomomorph-containing subgroup if, for any subgroup $$H$$ of $$N$$ and any homomorphism of groups $$\varphi:H \to G$$ such that $$\varphi(H)$$ is a normal subgroup of $$G$$, $$\varphi(H)$$ is contained in $$N$$.

Stronger properties

 * Weaker than::Subhomomorph-containing subgroup

Weaker properties

 * Stronger than::Normal-homomorph-containing subgroup
 * Stronger than::Weakly normal-homomorph-containing subgroup
 * Stronger than::Strictly characteristic subgroup
 * Stronger than::Characteristic subgroup