Group property-conditionally strongly 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 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 $$K$$.