Group property-conditionally image-potentially characteristic subgroup

Definition

Suppose $\alpha$ is a group property, $G$ is a group satisfying $\alpha$, and $H$ is a subgroup of $G$. We say that $H$ is an image-potentially characteristic subgroup relative to $\alpha$ if there exists a surjective homomorphism of groups $\rho :K\to G$ and a characteristic subgroup $L$ of $K$ such that $\rho (L)=H$.