Group property-conditionally strongly image-potentially characteristic subgroup

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