Weakly image-closed characteristic subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a weakly image-closed characteristic subgroup of $$G$$ if, for any normal subgroup $$N$$ of $$G$$ contained in $$H$$, the quotient group $$H/N$$ is a characteristic subgroup of the quotient group $$G/N$$.

Stronger properties

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

Weaker properties

 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup