Join-transiter of Abelian is central

Statement

Statement with symbols

Suppose $H$ is a subgroup of $G$ such that whenever $K$ is an Abelian subgroup (?) of $G$, the join of subgroups $\langle H,K\rangle$ is also an Abelian subgroup. Then, $H$ is a 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