Let $G$ be a (finite, say) group and $\alpha \in \mathrm{H}^3(\mathrm{B}G; \mathrm{U}(1))$ a 3-cohomology class. For each oriented 3-manifold $X^3$ equipped with a $G$-bundle $P : X \to \mathrm{B}G$, one can pull back $\alpha$ and integrate: $\int_X P^*\alpha \in \mathrm{U}(1)$.
I would like to restrict attention just to genus-one mapping tori. By this I mean: given $\gamma \in \mathrm{SL}(2,\mathbb Z)$, build the manifold $T(\gamma) = T^2 \times [0,1] /\sim$ where for $(x,y) \in T^2$, I set $((x,y),0) \sim (\gamma(x,y),1)$.
Question: Is the class $\alpha$ necessarily determined by the values of $\int_{T(\gamma)} P^*\alpha$ for all $\gamma \in \mathrm{SL}(2,\mathbb Z)$ and $P : T(\gamma) \to \mathrm{B}G$?