The purpose of this paper is to study finite simple groups whose Sylow 2-subgroups are isomorphic to those of PSL(5, q) for some odd q. This work originally appeared in the author's Ph.D. Thesis, (14), and at that time, the structure of the centralizers of involutions was worked out – see below for details. Subsequently, in view of the fact that if q ≡ – 1 (mod 4), the Sylow 2-subgroup of PSL(5, q) has sectional 2-rank 4 (being, in fact, isomorphic to the wreath product of a semidihedral group by a cyclic group of order 2), the complete classification, at least for this case, was needed for Gorenstein and Harada's monumental work on sectional 2-rank at most 4, (8), and therefore Collins and Solomon completed the characterization of PSL(5, q) and PSU(5, q) (for all odd q) in a neat paper, (5). Combining our result with theirs, we are able to state the following Theorem.