Projective modules and involutions

Let G be a finite group, and let Ω:={t Gt2=1}. Then Ω is a G-set under conjugation. Let k be an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module kΩ is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to a real 2-block of defect zero occurs with multiplicity 1 as a direct summand of kΩ, establishes a bijection between the projective components of kΩ and the real 2-blocks of G of defect zero.