Normality is not transitive for any nontrivially satisfied extension-closed group property

Statement

Suppose $p$ is a group property such that there is a nontrivial group satisfying $p$band $p$ is closed under taking extensions (in other words, if a normal subgroup and its quotient group both satisfy $p$, so does the whole group). Then, there exists a group $G$ satisfying $p$, a normal subgroup $K$ of $G$ satisfying $p$, and a normal subgroup $H$ of $K$, also satisfying $p$, that is not normal in $G$.

In fact, for any nontrivial group $H$ satisfying $p$, we can find $K$ and $G$ so that the above occurs.

Related facts

• Normality is not transitive
• Conjunction of normality with any nontrivial finite-direct product-closed group property is not transitive
• Every nontrivial normal subgroup is potentially normal-and-not-2-subnormal