Topological rigidity of automorphism actions on nilmanifolds

Suppose $X_{1}, X_{2}$ are nilmanifolds and $\rho, \sigma$ are automorphism actions of a discrete group $\Gamma$ on $X_{1}$ and $X_{2}$ respectively. We show that if $(X_{1},\rho)$ and $(X_{2}, \sigma)$ satisfy certain additional conditions then every $\Gamma$-equivariant continuous map from $(X_{1},\rho)$ to $(X_{2},\sigma)$ is an affine map.