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

Statement

Statement in terms of individual groups

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 normal subgroup of $K$ but not a characteristic subgroup of $K$.

Related facts

• Normal not implies characteristic
• Normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups