On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every $2$-generated subgroup is cyclic. Alperin proved that if $p$ is an odd prime then all finite $p$-groups with the property are metabelian. Nevertheless, finite Alperin $2$-groups may fail to be metabelian. We prove that for each finite abelian group $H$ there exists a finite Alperin group $G$ for which $G''$ is isomorphic to $H$.