Yes! Assuming char. $\neq2$, your $\sigma$ acts as $\pm{\rm id}$ on a local generator of $\omega_X$. In particular, the action on $\omega_X^{\otimes2}$ and its global sections is trivial.

It is much less obvious, but still true, that automorphisms of order $3$ and $5$ also act trivially on global sections of $\omega_X^{\otimes2}$. Mukai and Ohashi exploit this in their recent analysis of automorphisms of Enriques surfaces.