  We point out a gap in the proof of the Davis--Januszkiewicz theorem on
cohomology of small covers of simple polytopes, and give a correction to this
proof. We use this theorem to compute explicitly the Betti numbers for a wide
class of two-fold coverings over small covers. We describe a series of examples
in this class and explain why this class turns out to be special in the context
of the general problem of computing the modulo two Betti numbers of two-fold
coverings over small covers.
