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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.

Maximum orders of cyclic and abelian extendable actions on surfaces