Weakly normal-homomorph-containing subgroup

Definition with symbols
A subgroup $$N$$ of a group $$G$$ is termed a weakly normal-homomorph-containing subgroup if $$N$$ is a defining ingredient::normal subgroup of $$G$$ and the following holds:

Suppose $$\varphi:N \to G$$ is a homomorphism of groups such that for any normal subgroup $$H$$ of $$G$$ contained in $$N$$, we have $$\varphi(H)$$ is normal in $$G$$. Then, $$\varphi(N) \le N$$.