Normal equals strongly image-potentially characteristic

Statement
The following are equivalent for a subgroup $$H$$ of a group $$G$$ :


 * 1) $$H$$ is a normal subgroup of $$G$$.
 * 2) $$H$$ is a strongly image-potentially characteristic subgroup of $$G$$ in the following sense: there exists a group $$K$$ and a surjective homomorphism $$\rho:K \to G$$ such that both the kernel of $$\rho$$ and $$\rho^{-1}(H)$$ are characteristic subgroups of $$G$$.

Related facts

 * NPC theorem
 * NRPC theorem
 * Finite NPC theorem
 * Finite NIPC theorem
 * Fact about amalgam-characteristic subgroups: finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic

Facts used

 * 1) uses::Characteristicity is centralizer-closed

Proof
, slight modification of NRPC theorem.