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We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category,where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category iscalled a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G,\Gamma)$-crossed action. We show that every $(G, \Gamma)$-crossed tensorcategory $\mathcal C$ gives rise to a tensor category $\mathcal C^{(G,\Gamma)}$ that fits into an exact sequence of tensor categories$\operatorname{Rep G} \to \mathcal C^{(G, \Gamma)} \to \mathcal C$. We alsodefine the notion of a $(G, \Gamma)$-braiding in a $(G, \Gamma)$-crossed tensorcategory, which is connected with certain set-theoretical solutions of theQYBE. This extends the notion of $G$-crossed braided tensor category due toTuraev. We show that if $\mathcal C$ is a $(G, \Gamma)$-crossed tensor categoryequipped with a $(G, \Gamma)$-braiding, then the tensor category $\mathcalC^{(G, \Gamma)}$ is a braided tensor category in a canonical way.

Crossed actions of matched pairs of groups on tensor categories