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

Statement

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

Related facts

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