On finite Alperin $2$-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each $2$-generated subgroup is cyclic. Alperin proved that if $p$ is an odd prime then all finite p-groups with this property are metabelian. The today's actual problem is the construction of examples of nonmetabelian finite Alperin $2$-groups. Note that the author had given some examples of finite Alperin $2$-groups with second commutants isomorphic to $Z_2$ and $Z_4$ and proved the existence of finite Alperin $2$-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin $2$-groups with abelian second commutants of however large rank.