Every nontrivial normal subgroup is potentially 2-subnormal-and-not-normal

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 2-subnormal subgroup of $$K$$ but not a normal subgroup of $$K$$.

About the lack of transitivity of normality

 * Normality is not transitive
 * Normality is not transitive in any nontrivial extension-closed subquasivariety of the quasivariety of groups
 * Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive
 * There exist subgroups of arbitrarily large subnormal depth

The relation with characteristic subgroups

 * Characteristic of normal implies normal, left transiter of normal is characteristic
 * Every nontrivial normal subgroup is potentially normal-and-not-characteristic