The Brauer group of $BG$ and gerbe structures of moduli spaces

We study the $μ_N$-gerbe of curves of genus $g$ with an order $N$ automorphism, and explore what corresponding $H^2$-cohomology classes the components of this stack can have. In particular, we look at curves whose quotients by the order $N$ automorphism are genus 0, and completely determine the Brauer classes of these gerbes. The key technical input is the calculation of the Brauer group of $BG$, for $G$ a smooth connected semisimple linear algebraic group.