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