Assume we are in characteristic $\neq2$. ThenHowever, your $\sigma$ acts as Rita, Torsten and Ru pointed out, my first ideas were too simple-minded. Although this makes the remarks by them somewhat unreadable, let me give the corrected answer:

So, let $\pm{\rm id}$ on X$be a local generator of complex Enriques surface and$\omega_X$. Since \sigma$ an involution. Let us denote by $\tilde{X}\to X$ the associated K3-cover, and let $\tau$ be the associated involution, i.e., $X=\tilde{X}/\langle \tau\rangle$.

Now, the automorphism group of $X$ in terms of $\tilde{X}$ is {\rm Aut}(X)={\rm Aut}(\tilde{X},\tau) := {} ( \psi\in{\rm Aut}(\tilde{X})| \psi\tau=\tau\psi ) / \langle\tau\rangle(typesetting the usual brackets does not seem to work?!). In particular, $\sigma$ lifts to an Enriques surfaceautomorphism $\tilde{\sigma}$ of $\tilde{X}$.

Clearly, we have $\omega_X^{\otimes2}\cong{\cal O}_X$\tilde{\sigma}^2\in\langle\tau\rangle$. I claim that$\tilde{\sigma}^2={\rm id}$. We may thus find local generators For otherwise, we would have$m_\alpha$\tilde{\sigma}^2=\tau$, and $\tilde{\sigma}$ would be an automorphism of order $\omega_X$ 4$. In this case, since$\tau$acts freely on some open cover$U_\alpha$\tilde{X}$, the same would be true for $\tilde{\sigma}$. However, a K3 surface cannot possess a fixed-point free automorphism of order $X$, s.th. 4$: the quotient$m_\alpha^{\otimes2}$glue to the unique global section S$ of $\omega_X^{\otimes 2}$X$by this automorphism would satisfy$\chi({{\mathcal O}_S})=1/2$, which is absurd.But thenThus,$\sigma$acts trivially \tilde{\sigma}$ is an involution on this global section (since it $\tilde{X}$.

Being an involution, $\tilde{\sigma}$ acts via multiplication by as $\pm{\rm 1}$ id}$on the$m_\alpha$), i.e.1$-dimensional vectorspace $H^0(\omega_{\tilde{X}})$. Since $H^0(\omega_{\tilde{X}})^{\otimes2}\to H^0(\omega_{\tilde{X}}^{\otimes 2})$ is onto, we conclude that $\tilde{\sigma}$ acts trivially on all global sections of $\omega_{\tilde{X}}^{\otimes2}$.

Now, $\tilde{\sigma}$ induces $\sigma$ on $X$, and global sections of $\omega_X^{\otimes2}$ (since the space pull back to global sections of $\omega_{\tilde{X}}^{\otimes2}$. Since $\tilde{\sigma}$ acts trivially on these, we conclude that $\sigma$ acts trivially on global sections is of $1$-dimensional).\omega_X^{\otimes2}$. 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. 3 added 56 characters in body Yes! Assume we are in characteristic$\neq2$. Then, your$\sigma$acts as$\pm{\rm id}$on a local generator of$\omega_X$. Since$X$is an Enriques surface, we have$\omega_X^{\otimes2}\cong{\cal O}_X$. We may thus find local generators$m_\alpha$of$\omega_X$on some open cover$U_\alpha$of$X$, s.th. the$m_\alpha^{\otimes2}$glue to the unique global section of$\omega_X^{\otimes 2}$. But then,$\sigma$acts trivially on this global section (since it acts via multiplication by$\pm{\rm 1}$on$m_\alpha$), i.e., on all global sections of$\omega_X^{\otimes2}$.\omega_X^{\otimes2}$ (since the space of global sections is $1$-dimensional).

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.

Assume we are in characteristic $\neq2$, \neq2$. Then, your$\sigma$acts as$\pm{\rm id}$on a local generator of$\omega_X$. In particularSince$X$is an Enriques surface, we have$\omega_X^{\otimes2}\cong{\cal O}_X$. We may thus find local generators$m_\alpha$of$\omega_X$on some open cover$U_\alpha$of$X$, s.th. the action$m_\alpha^{\otimes2}$glue to the unique global section of$\omega_X^{\otimes 2}$. But then,$\sigma$acts trivially on this global section (since it acts via multiplication by$\omega_X^{\otimes2}$and its \pm{\rm 1}$ on $m_\alpha$), i.e., on all global sections is trivial.of $\omega_X^{\otimes2}$.
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.