  S.Bauer and M.Furuta defined a stable cohomotopy refinement of the
Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of
Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an
application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and
construct a nonsmoothable locally linear Z/2-action on K3#K3. We also construct
a nonsmoothable locally linear Z/2-action on $K3$.
