  Let $\Sigma_g (g>1)$ be a closed surface embedded in $S^3$. If a group $G$
can acts on the pair $(S^3, \Sigma_g)$, then we call such a group action on
$\Sigma_g$ extendable over $S^3$.
  In this paper we show that the maximum order of extendable cyclic group
actions is $4g+4$ when $g$ is even and $4g-4$ when $g$ is odd; the maximum
order of extendable abelian group actions is $4g+4$.
  We also give results of similar questions about extendable group actions over
handlebodies.
