The structure of cocycles of pseudo-homeomorphism groups

Let $\Gamma$ be a countable group acting ergodically as pseudo-homeomorphisms on a perfect Polish space $X$. It is proved that, modulo a meagre subset of $X$, any two ergodic cocycles $\alpha$ and $\beta$ of this action with values in a countable group $G$ are weakly equivalent. This result further applied to prove the outer conjugacy of a countable groups of pseudo-homeomorphisms from the normalizer $N[\Gamma]$ of a full group $[\Gamma]$.