  This article grew out of an effort to understand the smooth mapping class
groups of certain 4-manifolds in a geometric manner. We prove a smooth analog
of the Birman-Hilden theorem for manifolds that admit a hyperk\"ahler
structure. This allows us to probe the smooth mapping class groups associated
to certain manifolds with nontrivial fundamental groups. Along the way, and of
independent interest, we prove a global Torelli theorem for generalized
Enriques manifolds. The techniques used are analogous to
Teichm\"uller-theoretic methods in the classical theory of mapping class
groups. We apply this hyperk\"ahler Birman-Hilden theorem to obtain results
regarding smooth, metric, and complex Nielsen realization on Enriques surfaces.
