Normal subgroup having no nontrivial homomorphism to its quotient group

Definition
A subgroup $$N$$ of a group $$G$$ is a normal subgroup having no nontrivial homomorphism to its quotient group if $$N$$ is a normal subgroup of $$G$$ and there is no nontrivial homomorphism from $$N$$ to its quotient group $$G/N$$.

Related properties

 * Normal subgroup having no nontrivial homomorphism from its quotient group
 * Normal subgroup having no common composition factor with its quotient group

Other incomparable properties

 * Complemented normal subgroup:

Metaproperties
If $$H \le K \le G$$, with $$H$$ normal in $$G$$ and no nontrivial homomorphism from $$H$$ to $$G/H$$, $$H$$ is also normal in $$K$$ with no nontrivial homomorphism from $$H$$ to $$K/H$$.