Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariant

Statement

Suppose $G$ is a group and $H$ is a nontrivial Normal subgroup (?) of $G$. Then, there exists a group $K$ containing $G$ such that $H$ is a Characteristic subgroup (?) of $K$ but not a Fully invariant subgroup (?) of $K$.

Related facts

• Every nontrivial characteristic subgroup is potentially characteristic-and-not-fully invariant
• Characteristic not implies fully invariant
• NPC theorem: This states that every normal subgroup can be realized as a characteristic subgroup in some bigger group.
• Normal not implies potentially fully invariant