Image-potentially fully invariant subgroup

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is an image-potentially fully invariant subgroup of $$G$$ if there exists a group $$K$$, a surjective homomorphism $$\rho:K \to G$$, and a subgroup $$L$$ of $$K$$ such that $$\rho(L) = H$$.