Fixed point sets for permutation modules

We investigate the representation of the symmetric group afforded by the action on its conjugacy class of fixed point free involutions, over an algebraically closed field of finite characteristic p. We discuss the general form of the set of involutions fixed by an arbitrary vertex of an irreducible component of the module afforded by the action, and show how such a set is composed of certain basic fixed point sets which we can enumerate in the case p = 2. The Brou\'e correspondence allows us to pass from the fixed point sets to the vertices.