  We present examples of prequantizations over integral symplectic manifolds
which admit infinitely many smoothly trivial contact mapping classes. These
classes are given by the connected components of the strict contactomorphism
group which project to the identity component of the symplectomorphism group of
the base manifold. Along the way, we study the lifting problem of
symplectomorphisms of the base manifold to strict contactomorphisms of the
prequantization.
