  We first show that every group-theoretical category is graded by a certain
double coset ring. As a consequence, we obtain a necessary and sufficient
condition for a group-theoretical category to be nilpotent. We then give an
explicit description of the simple objects in a group-theoretical category
(following Ostrik, arXiv:math/0202130) and of the group of invertible objects
of a group-theoretical category, in group-theoretical terms. Finally, under
certain restrictive conditions, we describe the universal grading group of a
group-theoretical category.
