Central factor is not finite-join-closed

Statement with symbols
It is possible to have a finite group $$G$$ with subgroups $$H, K$$ such that both $$H$$ and $$K$$ are central factors of $$G$$ but the product $$HK$$ (which in this case is also the join of subgroups $$\langle H, K \rangle$$) is not a central factor.

Central factor
A subgroup $$H$$ of a group $$G$$ is termed a central factor if $$HC_G(H) = G$$, or equivalently, every inner automorphism of $$G$$ restricts to an inner automorphism of $$H$$.

Example of the extraspecial group
For any prime $$p$$, either of the two isomorphism classes of extraspecial groups of order $$p^5$$ gives a counterexample.