  We give an explicit description of the Godeaux surfaces that admit an
involution such that the quotient surface is birational to an Enriques surface;
these surfaces give a 6-dimensional unirational irreducible subset of the
moduli space of surfaces of general type. In addition, we describe the Enriques
surfaces that are birational to the quotient of a Godeaux surface by an
involution and we show that they give a 5-dimensional unirational irreducible
subset of the moduli space of Enriques surfaces. Finally, by degenerating our
description we obtain some examples of non-normal stable Godeaux surfaces.
