The Parity of Invariant Characteristics

We demonstrate a method to prove that a theta characteristic on Riemann surface $S$ invariant under the action of $G\le \mathrm{A}\mathrm{u}\mathrm{t}(S)$ has even parity by considering the representation theory of the double cover $\tilde{G}=2\cdot G$. We apply this to certain Hurwitz curves, and prove a recent conjecture of Broughton & Disney-Hogg.