  An automorphism of an algebraic surface $S$ is called cohomologically
(numerically) trivial if it acts identically on the second $l$-adic cohomology
group (this group modulo torsion subgroup). Extending the results of S. Mukai
and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of
cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques
surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and
$S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of
odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order $8$
and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also
prove that the group $\rm{Aut}_{\rm{nt}}(S)$ of numerically trivial
automorphisms is a subgroup of a cyclic group of order $\leq 4$ unless $p = 2$,
where $\rm{Aut}_{\rm{nt}}(S)$ is a subgroup of a $2$-elementary group of rank
$\leq 2$.
