Extendable periodic automorphisms of closed surfaces over the 3-sphere

A periodic automorphism of a surface $\mathrm{\Sigma }$ is said to be extendable over $S^3$ if it extends to a periodic automorphism of the pair $(S^3,\mathrm{\Sigma })$ for some possible embedding $\mathrm{\Sigma }\to S^3$. We classify and construct all extendable automorphisms of closed surfaces, with orientation-reversing cases included. Moreover, they can all be induced by automorphisms of $S^3$ on Heegaard surfaces. As a by-product, the embeddings of surfaces into lens spaces are discussed.