  We explore the category of internal categories in the usual category of
(right) group-sets, whose objects are referred to as categorified group-sets.
More precisely, we develop a new Burnside theory, where the equivalence
relation between two categorified group-sets is given by a particular
equivalence between the underlying categories. We also exhibit some of the
differences between the old Burnside theory and the new one. Lastly, we briefly
explain how to extend these new techniques and concepts to the context of
groupoids, employing the categories of (right) groupoid-sets, aiming by this to
give an alternative approach to the classical Burnside ring of groupoids.
