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We develop a categorical analogue over graded fusion categories of Cliffordtheory for strongly graded rings. We describe module categories over a fusioncategory graded by a group $G$ as induced from module categories over fusionsubcategories associated with the subgroups of $G$. We define invariant$\C_e$-module categories and extensions of $\C_e$-module categories. Theconstruction of module categories over $\C$ is reduced to determine invariantmodule categories for subgroups of $G$ and the indecomposable extensions ofthis modules categories. We associate a $G$-crossed product fusion category toeach $G$-invariant $\C_e$-module category and give a criterion for a gradedfusion category to be a group-theoretical fusion category. We give necessaryand sufficient conditions for an indecomposable module category to be extended.

Clifford theory for graded fusion categories