Embedding periodic maps on surfaces into those on S3

Call a periodic map h on the closed orientable surface Σ g extendable if h extends to a periodic map over the pair ( S 3 ,Σ g ) for possible embeddings e : Σ g → S 3 . The authors determine the extendabilities for all periodical maps on Σ 2 . The results involve various orientation preserving/reversing behalves of the periodical maps on the pair ( S 3 ,Σ g ). To do this the authors first list all periodic maps on Σ 2 , and indeed the authors exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be interesting. A by-product is that for each even g , the maximum order periodic map on Σ g is extendable, which contrasts sharply with the situation in the orientation preserving category.