Join-transiter of Abelian is central

Statement with symbols
Suppose $$H$$ is a subgroup of $$G$$ such that whenever $$K$$ is an fact about::Abelian subgroup of $$G$$, the join of subgroups $$\langle H,K \rangle$$ is also an Abelian subgroup. Then, $$H$$ is a fact about::central subgroup of $$G$$: $$H$$ is contained in the center of $$G$$.

Related facts

 * Join of Abelian and central implies Abelian
 * Cyclic over central implies Abelian