Finite groups with biprimary subgroups of a definite form

The paper studies the structure of finite groups in which, for any biprimary subgroup $B$, either $l_2(B)\le1$ or $O_2(B)$ is a metacyclic group. As a corollary of the result obtained here and of known results of other authors, a description is adduced of finite simple groups in which the intersection of any two distinct Sylow 2-subgroups is metacyclic.