Finite groups in which a Sylow two-subgroup of the centralizer of some involution is of order 16

It is proved that the sectional two-rank of a finite group $G$ having no subgroup of index two is at most four if a Sylow two-subgroup of the centralizer of some involution of $G$ is of order 16. This implies the following assertion: If $G$ is a finite simple group whose order is divisible by $2^5$ and the order of the centralizer of some involution of $G$ is not divisible by $2^5$, then $G$ is isomorphic to the Mathieu group $M_{12}$ or the Hall–Janko group $J_2$.