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

Statement in terms of individual groups
Suppose $$G$$ is a group and $$H$$ is a nontrivial fact about::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