We study how Weierstrass points of Veech surfaces in H (2), the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two, are permuted. These surfaces were classified by McMullen relying on two invariants: discriminant and spin. More precisely, given a Veech surface in H (2) of discriminant D, we show that the permutation group induced by the affine group on the set of Weierstrass points is isomorphic to Dih4, if D equivalent to 4 0; to Dih5, if D equivalent to 8 5; and to Dih6, if D equivalent to 8 1. Moreover, these same groups arise when considering only Dehn multitwists of the affine group.