Central subgroup implies join-transitively central factor

Statement with symbols
Suppose $$G$$ is a group and $$H$$ is a central subgroup of $$G$$, i.e., $$H$$ is contained in the center of $$G$$. Suppose $$K$$ is a central factor of $$G$$. Then, the join of subgroups $$\langle H, K \rangle$$, which is also the product of subgroups $$HK$$, is a central factor of $$G$$.

Related facts

 * Cocentral implies join-transitively central factor
 * Right-quotient-transitively central factor implies join-transitively central factor
 * Direct factor implies join-transitively central factor
 * Central factor is not finite-join-closed