Clifford theory for graded fusion categories

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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 ₑ-module categories and extensions of ₑ-module categories. The construction of module categories over is reduced to determine invariant module categories for subgroups of G and the indecomposable extensions of this modules categories. We associate a G-crossed product fusion category to each G-invariant 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 extended.