Strongly image-potentially characteristic subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a strongly image-potentially characteristic subgroup if there exists a surjective homomorphism $$\rho:K \to G$$ such that both the kernel of $$\rho$$ and $$\rho^{-1}(H)$$ are characteristic subgroups of $$K$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Kernel of a characteristic action on an abelian group

Weaker properties

 * Stronger than::Semi-strongly image-potentially characteristic subgroup
 * Stronger than::Image-potentially characteristic subgroup
 * Stronger than::Normal subgroup

Facts

 * Finite NIPC theorem: This shows that any normal subgroup of a finite group is strongly image-potentially characteristic.
 * No nontrivial abelian normal p-subgroup for some prime p implies every normal subgroup is strongly image-potentially characteristic: This shows that for a very large class of groups, every normal subgroup is strongly image-potentially characteristic.