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For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the "abelian cohomology of G-modules". Then, as the main results of this paper, natural one-toone correspondences between elements of the 3 rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all thesedoi:10.1016/j.jpaa.2006.06.011 fatcat:lvsnfcnyefb4fj4gqvq4v4f5he