Let G be a �nite group. In this paper we investigate the permutation module of G acting by conjugation on its involutions, over a �eld of characteristic 2. This develops the main theme of [10] and [8]. In the former paper G. R. Robinson considered the projective components of this module. In the latter paper the author showed that each such component is irreducible and self-dual and belongs to a 2-blocks of defect zero. Here we investigate which 2-blocks have a composition factor in the involution module. There are two apparently di�erent ways of characterising such blocks. One method is local and uses the defect classes of the block. This gives rise to the de�nition of a strongly real 2-block. The other method is global and uses the Frobenius-Schur indicators of the irreducible characters in the block. Our main result is Theorem 2. The proof of this theorem requires Corollaries 4, 15, 18 and 20.