  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.
