Embedding periodic maps on surfaces into those on $S^3$

Call a periodic map $h$ on the closed orientable surface $\Sigma_g$ extendable if $h$ extends to a periodic map over the pair $(S^3, \Sigma_g)$ for possible embeddings $e: \Sigma_g\to S^3$. We determine the extendabilities for all periodical maps on $\Sigma_2$. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair $(S^3, \Sigma_g)$. To do this we first list all periodic maps on $\Sigma_2$, and indeed we exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be an interesting piece. A by-product is that for each even $g$, the maximum order periodic map on $\Sigma_g$ is extendable, which contrasts sharply to the situation in orientation preserving category.