Group property-conditionally semi-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 semi-strongly image-potentially characteristic in $$G$$ relative to $$\alpha$$ if there exists a group $$K$$ satisfying $$\alpha$$ and a surjective homomorphism $$\rho:K \to G$$ such that $$\rho^{-1}(H)$$ is a characteristic subgroup of $$K$$.