The shift action on the 2-cocycle group Z(2)(G, C) of a finite group G with coefficients in a finitely generated abelian group C has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action. In this article, we study the shift orbit structure of the coboundary subgroup B-2 (G, C) of Z(2) (G, C). The study is placed within a well-known setting involving the Loewy and socle series of a group algebra over G. We prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if C is an elementary abelian p-group, then almost all shift orbits in B-2 (G, C) are maximal-sized for large enough finite p-groups G of certain classes. (C) 2009 Elsevier B.V. All rights reserved.