Potentially normal-subhomomorph-containing equals normal

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 fact about::normal-subhomomorph-containing subgroup of $$K$$.
 * 3) There exists a group $$K$$ containing $$G$$ such that $$H$$ is a fact about::normal-homomorph-containing subgroup of $$K$$.
 * 4) There exists a group $$K$$ containing $$G$$ such that $$H$$ is a fact about::strictly characteristic subgroup of $$K$$.

Proof
The same construction as used for the NPC theorem works.