For an abelian group $A$, we study a close connection between braided $A$-crossed tensor categories with trivialization of the $A$-action and $A$-graded categories. Additionally, prove that obstruction to existence categorical action $T$ on category $\mathcal{C}$ is given by element $O(T) \in H^2(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$. In case = 0$, set obstructions ...
