Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive

Statement
Suppose $$p$$ is a group property that is finite-direct product-closed and there is at least one nontrivial group satisfying $$p$$. Then, it is possible to have groups $$H \le K \le G$$ such that $$K$$ is normal in $$G$$, $$H$$ is normal in $$K$$, both $$H$$ and $$K$$ satisfy $$p$$, and $$H$$ is not a fact about::normal subgroup of $$G$$.

Related facts

 * Stronger than::Normality is not transitive
 * Normality is not transitive in any nontrivial extension-closed subquasivariety of the quasivariety of groups