The Krull-Schmidt-Remak-Azumaya Theorem for G-groups

We present some results related to G-groups and their category G-Grp in connection with the category Grp of all groups and the category R-Mod of all left modules over a fixed ring R. In particular, we prove that the Krull-Schmidt-Remak-Azumaya Theorem holds for G-groups, where G is a fixed group, for finite families of G-groups. A key role in this setting is played by the class of completely indecomposable G-groups.