  Let $F_g$ be the closed orientable surface of genus $g$. We address the
problem to extend torsion elements of the mapping class group
${\mathcal{M}}(F_g)$ over the 4-sphere $S^4$. Let $w_g$ be a torsion element of
maximum order in ${\mathcal{M}}(F_g)$. Results including:
  (1) For each $g$, $w_g$ is periodically extendable over $S^4$ for some
non-smooth embedding $e: F_g\to S^4$, and not periodically extendable over
$S^4$ for any smooth embedding $e: F_g\to S^4$.
  (2) For each $g$, $w_g$ is extendable over $S^4$ for some smooth embedding
$e: F_g\to S^4$ if and only if $g=4k, 4k+3$.
  (3) Each torsion element of order $p$ in ${\mathcal{M}}(F_g)$ is extendable
over $S^4$ for some smooth embedding $e: F_g\to S^4$ if
  either (i) $p=3^m$ and $g$ is even;
  or (ii) $p=5^m$ and $g\ne 4k+2$;
  or (iii) $p=7^m$.
  Moreover the conditions on $g$ in (i) and (ii) can not be removed .
