An equivariant BGG correspondence and perfect complexes for extensions by Z/2× Z/2

We provide an equivariant extension of Carlsson's BGG correspondence in characteristic two. As an application we classify perfect cochain complexes of (Z/2× Z/2) Q-representations with four-dimensional total homology for finite groups Q of odd order. We deduce that cochain complexes of finite, free A₄-CW complexes with four-dimensional total homology are rigid: They are determined by the degrees of the nonzero homology groups.