Godeaux surfaces with an Enriques involution and some stable degenerations

We give an explicit description of the Godeaux surfaces (minimal surfaces of general type with $K^2_S=1$, $p_g=0$ ) that admit an involution such that the quotient surface is birational to an Enriques surface; these surfaces give a six-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 five-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; in particular we show that one of the families of stable Gorenstein Godeaux surfaces classified in [Franciosi, Pardini, Rollenske, Log-canonical pairs and Gorenstein stable surfaces with K^2_X=1, Comp. Math, vol. 151, p. 1529-1542 (2015)] consists of smoothable surfaces.