  We give a list of possibilities for surfaces of general type with $p_g=0$
having an involution $i$ such that the bicanonical map of $S$ is not composed
with $i$ and $S/i$ is not rational. Some examples with $K^2=4,...,7$ are
constructed as double coverings of an Enriques surface. These surfaces have a
description as bidouble coverings of the plane.
