Permutation of periodic points of Veech surfaces in $\mathcal{H}(2)$

We study how are permuted Weierstrass points of Veech surfaces in $\mathcal{H}(2)$, the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two. These surfaces were classified by McMullen relying on two invariants: discriminant and spin. More precisely, given a Veech surface in $\mathcal{H}(2)$ of discriminant $D$, we show that the permutation group induced by the affine group on the set of Weierstrass points is isomorphic to $\mathrm{Dih}_4$, if $D \mathbin{\equiv_{4}} 0$; to $\mathrm{Dih}_5$, if $D \mathbin{\equiv_{8}} 5$; and to $\mathrm{Dih}_6$, if $D \mathbin{\equiv_{8}} 1$. Moreover, these same groups arise when considering only Dehn multitwists of the affine group.