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We classify the bi-canonical representations of finite automorphisms onEnriques surfaces. There are three types of non-trivial cases and examples aregiven explicitly by Horikawa models. In particular, finite non-semi-symplecticautomorphisms exist only in orders 4 and 8. One corollary is that any finitecyclic subgroup in the automorphism group of an Enriques surface has order 1,2, 3, 4, 5, 6, 8. Moreover, for two of the three types, a uniqueness theoremfor maximal-dimensional families is given.

Bi-canonical representations of finite automorphisms acting on Enriques surfaces