  Given a group cocycle on a finitely aligned left cancellative small category
(LCSC) we investigate the associated skew product category and its
Cuntz-Krieger algebra, which we describe as the crossed product of the
Cuntz-Krieger algebra of the original category by an induced coaction of the
group. We use our results to study Cuntz-Krieger algebras arising from free
actions of groups on finitely aligned LCSC's, and to construct coactions of
groups on Exel-Pardo algebras. Finally we discuss the universal group of a
small category and connectedness of skew product categories.
