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The aim of this article is to classify the pairs (S, G), where S is a smoothminimal surface of general type with p_g=0 and K^2=7, G is a subgroup of theautomorphism group of S and G is isomorphic to the group $\mathbb{Z}_2^2$. TheInoue surfaces with K^2=7, which are finite Galois $\mathbb{Z}_2^2$-covers ofthe 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. Weprove that the base of the Kuranishi family of deformations of a surface inthis family is smooth. We show that, in the Gieseker moduli space of canonicalmodels of surfaces of general type, the subset corresponding to the surfaces inthis family is an irreducible connected component, normal, unirational ofdimension 3.

Commuting involutions on surfaces of general type with p_g=0 and K^2=7