Semi-strongly image-potentially characteristic subgroup

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

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Kernel of a characteristic action on an abelian group
 * Weaker than::Strongly image-potentially characteristic subgroup

Weaker properties

 * Stronger than::Image-potentially characteristic subgroup
 * Stronger than::Normal subgroup