Central 2-sylow intersections

Let G be a finite group. A subgroup Z) of G is called a 2-Sylow intersection if there exist distinct Sylow 2-subgroups Si and S2 of G such that D = S₁ Ç S₂. An involution of G is called central if it is contained in a center of a Sylow 2-subgroup of G. A 2-Sylow intersection is called central if it contains a central involution. The aim of this work is to determine all non-abelian simple groups G which satisfy the following condition B: The 2-rank of all central 2-Sylow intersections is not higher than 1, under the additional assumption that the cen­tralizer of a central involution of G is solvable.