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 $K$.

Related facts

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