Image-closed characteristic subgroup

Symbol-free definition
A subgroup of a group is termed an image-closed characteristic subgroup if, under any surjective homomorphism, its image is a characteristic subgroup of the image.

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

Stronger properties

 * Weaker than::Verbal subgroup
 * Weaker than::Image-closed fully invariant subgroup

Weaker properties

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

Related properties

 * Intermediately characteristic subgroup