Clifford theory for graded fusion categories

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group G as induced from module categories over fusion subcategories associated with the subgroups of G. We define invariant C _e -module categories and extensions of C _e -module categories. The construction of module categories over C is reduced to determining invariant module categories for subgroups of G and the indecomposable extensions of these module categories. We associate a G-crossed product fusion category to each G-invariant C _e -module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.