Numerically trivial automorphisms of Enriques surfaces in characteristic $2$

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 ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{c}\mathrm{t}}(S)$ of an Enriques surface $S$ is of order $\le 2$ if $S$ is not supersingular. If $p=2$ and $S$ is supersingular, we show that ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{c}\mathrm{t}}(S)$ 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 $K_S\mathrm{\ne }0$, we also prove that the group ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{n}\mathrm{t}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\le 4$ unless $p=2$, where ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{n}\mathrm{t}}(S)$ is a subgroup of a $2$-elementary group of rank $\le 2$.