Bundles with multivalued automorphism groups

A category whose objects are principal bundles with fixed base (smooth manifold) $B$, structure group $T^k$, and finite group $\Delta$ of multivalued automorphisms is constructed; the morphisms are required to be equivariant with respect to $\Delta$. Invariants are found and used to calculate the group of equivalence classes of the category objects. Examples are given and applications to dynamical systems with gyroscopic forces are suggested.