  We prove that every dynamical system $X$ with free action of a countable
amenable group $G$ by homeomorphisms has a zero-dimensional extension $Y$ which
is faithful and principal, i.e. every $G$-invariant measure $\mu$ on $X$ has
exactly one preimage $\nu$ on $Y$ and the conditional entropy of $\nu$ with
respect to $X$ is zero. This is a version of an earlier result by T.
Downarowicz and D. Huczek, which establishes the existence of zero-dimensional
principal and faithful extensions for general actions of the group of integers.
