Some Simple Projective Brauer Quotients of Simple Modules for the Symmetric Groups in Characteristic Two

By Alperinʹs weight conjecture (J. L. Alperin and P. Fong, 1990, J. Algebra131, 2–22) the number of simple kSn-modules equals the number of weights for Sn, where Sn is the symmetric group on n symbols and k is a field of characteristic p > 0. In this paper we answer the question, “When is the Brauer quotient of a simple F2Sn-module V with respect to a subgroup H of Sn both simple and projective as an NSn(H)/H-module?,” in some special cases. Remarkably, in each case there is only one such subgroup H (up to conjugacy).