Weakly image-closed fully invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a weakly image-closed fully invariant subgroup if, for any normal subgroup $$N$$ of $$G$$ contained in $$H$$, $$H/N$$ is a defining ingredient::fully invariant subgroup of $$G/N$$.

Stronger properties

 * Weaker than::Image-closed fully invariant subgroup
 * Weaker than::Quotient-subisomorph-containing subgroup
 * Weaker than::Verbal subgroup

Weaker properties

 * Stronger than::Fully invariant subgroup
 * Stronger than::Weakly image-closed characteristic subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup