Pronormality is not permuting join-closed

Statement
It is possible to have a group $$G$$ and two permuting subgroups $$H, K$$ of $$G$$ such that both $$H$$ and $$K$$ are pronormal subgroups but the join $$\langle H, K \rangle$$, which in this case equals the product $$HK$$, is not pronormal.

Related facts about pronormality

 * Pronormality is normalizing join-closed
 * Pronormality is not finite-join-closed
 * Pronormality is not commutator-closed
 * Pronormality is not centralizer-closed

Related facts about permuting joins

 * Subnormality is permuting join-closed
 * Subnormality is normalizing join-closed