  For an abelian group $ A $, we study a close connection between braided
crossed $ A $-categories with a trivialization of the $ A $-action and $ A
$-graded braided tensor categories. Additionally, we prove that the obstruction
to the existence of a trivialization of a categorical group action $T$ on a
monoidal category $\mathcal{C}$ is given by an element $O(T)\in
H^2(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$. In the
case that $O(T)=0$, the set of obstructions form a torsor over
$\operatorname{Hom}(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$,
where $\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}})$ is the
abelian group of tensor natural automorphisms of the identity.
  The cohomological interpretation of trivializations, together with the
homotopical classification of (faithfully graded) braided $A$-crossed tensor
categories developed in arXiv:0909.3140, allows us to provide a method for the
construction of faithfully $A$-graded braided tensor categories. We work out
two examples. First, we compute the obstruction to the existence of
trivializations for the braided crossed category associated with a pointed
semisimple tensor category. In the second example, we compute explicit formulas
for the braided $\mathbb{Z}/2$-crossed structures over Tambara-Yamagami fusion
categories and, consequently, a conceptual interpretation of the results in
arXiv:math/0011037 about the classification of braidings over Tambara-Yamagami
categories.
