The authors study regular automorphisms of oriented maps. Their main concern is the problem of lifting and projecting map automorphisms. Regular homomorphisms of oriented maps essentially arise from a factorization by a subgroup of automorphisms. In this paper, this kind of map automorphisms is studied in detail. Moreover, the results are generalized to the case when the induced homomorphism of the underlying graph is not valency preserving. Reconstruction is treated by means of voltage assignment on angles. Lifting and projecting maps are studied. The split-extension structure of lifted groups is analysed. In particular, those homomorphisms \(\widetilde M\to M\) between regular maps for which \(\text{Aut }\widetilde M\) is a split extension of the fibre transformation group by \(\text{Aut }M\) are characterized.