We show that superselection structures on curved spacetimes, that are expected to describe quantum charges affected by the underlying geometry, are categories of sections of presheaves of symmetric tensor categories. This implies that, provided an embedding functor (whose existence and uniqueness are not guaranteed), the superselection structure is a Tannaka-type dual of a locally constant group bundle, which hence becomes a natural candidate for the role of gauge group. Indeed, we show that any locally constant group bundle (with suitable structure group) acts on a net of C*-algebras fulfilling normal commutation relations on an arbitrary spacetime. We also give examples of gerbes of C*-algebras, defined by Wightman fields and constructed using projective representations of the fundamental group of the spacetime.