Potentially normal-subhomomorph-containing equals normal

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

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

1. $H$ is a normal subgroup of $G$.
2. There exists a group $K$ containing $G$ such that $H$ is a Normal-subhomomorph-containing subgroup (?) of $K$.
3. There exists a group $K$ containing $G$ such that $H$ is a Normal-homomorph-containing subgroup (?) of $K$.
4. There exists a group $K$ containing $G$ such that $H$ is a Strictly characteristic subgroup (?) of $K$.

Proof

The same construction as used for the NPC theorem works.