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 $$\rho(L) = H$$.