Neutral representations of finite diagonalizable group schemes and fields of moduli

We introduce the notion of a neutral representation of a finite group, or finite group scheme, $G$; a representation $V$ with the property that if a gerbe $\mathcal{G}$ over a field $k$ that is a form of the classifying stack $\mathcal{B}G$ admits a vector bundle that is a form of $V$, then it is neutral, that is, $\mathcal{G}(k)$ is not empty. We give some criteria for a representation of a finite diagonalizable group scheme to be neutral. We apply this notion to give wide classes of examples of smooth curves, or varieties with a marked point, with cyclic automorphism groups, which are defined over their field of moduli, greatly generalizing some previous results.