 homomorphism $\pi:B\to {\operatorname{\mathcal{NO}}_{\!X}}^{r}$ is injective if and only if there is a coaction $\beta$ of $G$ on $B$ such that $\pi$ is $\beta-\nu^n$ equivariant and $\pi\vert_{B_e}$ is injective.  Apply Corollary \[cor: