Let G be a finite group, let k be an algebraically closed field of characteristic 2 and let Omega := {g is an element of G vertical bar g(2) = 1(G)}. It is shown that for a block B of kG, the permutation module k Omega has a B-composition factor if and only if the Frobenius-Schur indicator of the regular character of B is non-zero or equivalently if and only if B is real with a strongly real defect class.