  In this paper, we prove a general principle of lifting an automorphism from
positive characteristic to zero characteristic. We based on the principle to
prove the automorphism group of Fano variety of cubic threefold (fourfold) acts
on its cohomology group faithfully. We also prove the same result for a
singular cubic threefold (fourfold) with one ordinary double singularity. As an
application, we classify the automorphism groups of cubic fourfolds with small
middle Picard number over complex numbers.
