  We develop a method for generating the complete set of basic data under the
torsorial actions of $H^2_{[\rho]}(G,\mathcal{A})$ and $H^3(G,\text{U}(1))$ on
a $G$-crossed braided tensor category $\mathcal{C}_G^\times$, where
$\mathcal{A}$ is the set of invertible simple objects in the braided tensor
category $\mathcal{C}$. When $\mathcal{C}$ is a modular tensor category, the
$H^2_{[\rho]}(G,\mathcal{A})$ and $H^3(G,\text{U}(1))$ torsorial action gives a
complete generation of possible $G$-crossed extensions, and hence provides a
classification. This torsorial classification can be (partially) collapsed by
relabeling equivalences that appear when computing the set of $G$-crossed
braided extensions of $\mathcal{C}$. The torsor method presented here reduces
these redundancies by systematizing relabelings by $\mathcal{A}$-valued
$1$-cochains. We also use our methods to compute the composition rule of these
torsor functors.
