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The aim of this article is to classify the pairs (S, G), where S is a smooth minimal surface of general type with p  g  =  0 and K 2 =  7, G is a subgroup of the automorphism group of S and G is isomorphic to the group \({{\mathbb{Z}}_2^2}\). We show that there are only three possible cases for such pairs. Two of them correspond to known examples, but the existence of the third one remains an open problem. The Inoue surfaces with K 2 =  7, which are finite Galois \({\mathbb{Z}_2^2}\)-covers of the 4-nodal cubic surface, are the first examples of such pairs. More recently, the author constructed a new family of such pairs. They are finite Galois \({\mathbb{Z}_2^2}\)-covers of certain 6-nodal Del Pezzo surfaces of degree one. We prove that the base of the Kuranishi family of deformations of a surface in this family is smooth. We show that, in the Gieseker moduli space of canonical models of surfaces of general type, the subset corresponding to the surfaces in this family is an irreducible connected component, normal, unirational of dimension 3.

Commuting involutions on surfaces of general type with p g = 0 and K 2 = 7