Categorical G-crossed modules and 2-fold extensions

In this paper we define the notions of categorical G-crossed modules for a group G and of 2-fold extensions of G by a symmetric G-categorical group image. These concepts cover, respectively, the usual notions of crossed modules of groups and of 2-fold extensions of a group G by a G-module A as well as those due to Conduché of 2-crossed modules and non-abelian 3-extensions of G by a G-module A. We prove that there exists a bijection between the set of equivalence classes of 2-fold extensions of G by image and image, the fourth Ulbrich (Fröhlich–Wall) cohomology group of G with coefficients in image.