Topological rigidity of automorphism actions on nilmanifolds

Suppose $X_1,X_2$ are nilmanifolds and $\rho ,\sigma$ are automorphism actions of a discrete group $\mathrm{\Gamma }$ on $X_1$ and $X_2$ respectively. We show that if $(X_1,\rho )$ and $(X_2,\sigma )$ satisfy certain additional conditions then every $\mathrm{\Gamma }$-equivariant continuous map from $(X_1,\rho )$ to $(X_2,\sigma )$ is an affine map.