  Let $X$ be an Enriques surface. Using Beauville's result about the triviality
of the Brauer map of $X$, we define a new involution on the category of
coherent sheaves on the canonically covering K3 surface $\overline{X}$. We
relate the fixed locus of this involution to certain Picard schemes of the
noncommutative pair $(X,\mathcal{A})$, where $\mathcal{A}$ is an Azumaya
algebra on $X$ defined by the nontrivial element in the Brauer group of $X$.
