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Yes! However, 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 $X$ be a complex Enriques surface and $\sigma$ 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 automorphism $\tilde{\sigma}$ of $\tilde{X}$.

Clearly, we have $\tilde{\sigma}^2\in\langle\tau\rangle$. I claim that $\tilde{\sigma}^2=\{\rm id}$. For otherwise, we would have $\tilde{\sigma}^2=\tau$, and $\tilde{\sigma}$ would be an automorphism of order $4$. In this case, since $\tau$ acts freely on $\tilde{X}$, the same would be true for $\tilde{\sigma}$. However, a K3 surface cannot possess a fixed-point free automorphism of order $4$: the quotient $S$ of $X$ by this automorphism would satisfy $\chi({{\mathcal O}_S})=1/2$, which is absurd. Thus, $\tilde{\sigma}$ is an involution on $\tilde{X}$.

Being an involution, $\tilde{\sigma}$ acts as $\pm{\rm id}$ on the $1$-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 global sections of $\omega_{\tilde{X}}^{\otimes2}$.

Now, $\tilde{\sigma}$ induces $\sigma$ on $X$, and global sections of $\omega_X^{\otimes2}$ 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 of $\omega_X^{\otimes2}$.

It is 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.

Yes! However, 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 $X$ be a complex Enriques surface and $\sigma$ 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 automorphism $\tilde{\sigma}$ of $\tilde{X}$.

Clearly, we have $\tilde{\sigma}^2\in\langle\tau\rangle$. I claim that $\tilde{\sigma}^2={\rm id}$. For otherwise, we would have $\tilde{\sigma}^2=\tau$, and $\tilde{\sigma}$ would be an automorphism of order $4$. In this case, since $\tau$ acts freely on $\tilde{X}$, the same would be true for $\tilde{\sigma}$. However, a K3 surface cannot possess a fixed-point free automorphism of order $4$: the quotient $S$ of $X$ by this automorphism would satisfy $\chi({{\mathcal O}_S})=1/2$, which is absurd. Thus, $\tilde{\sigma}$ is an involution on $\tilde{X}$.

Being an involution, $\tilde{\sigma}$ acts as $\pm{\rm id}$ on the $1$-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 global sections of $\omega_{\tilde{X}}^{\otimes2}$.

Now, $\tilde{\sigma}$ induces $\sigma$ on $X$, and global sections of $\omega_X^{\otimes2}$ 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 of $\omega_X^{\otimes2}$.

It is 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.

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 $\omega_X^{\otimes2}$ (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.

Yes! However, 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 $X$ be a complex Enriques surface and $\sigma$ 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 automorphism $\tilde{\sigma}$ of $\tilde{X}$.

Clearly, we have $\tilde{\sigma}^2\in\langle\tau\rangle$. I claim that $\tilde{\sigma}^2=\{\rm id}$. For otherwise, we would have $\tilde{\sigma}^2=\tau$, and $\tilde{\sigma}$ would be an automorphism of order $4$. In this case, since $\tau$ acts freely on $\tilde{X}$, the same would be true for $\tilde{\sigma}$. However, a K3 surface cannot possess a fixed-point free automorphism of order $4$: the quotient $S$ of $X$ by this automorphism would satisfy $\chi({{\mathcal O}_S})=1/2$, which is absurd. Thus, $\tilde{\sigma}$ is an involution on $\tilde{X}$.

Being an involution, $\tilde{\sigma}$ acts as $\pm{\rm id}$ on the $1$-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 global sections of $\omega_{\tilde{X}}^{\otimes2}$.

Now, $\tilde{\sigma}$ induces $\sigma$ on $X$, and global sections of $\omega_X^{\otimes2}$ 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 of $\omega_X^{\otimes2}$.

It is 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.

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 $\omega_X^{\otimes2}$.

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.

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 $\omega_X^{\otimes2}$ (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.