Extending finite group actions on surfaces over $S^3$

Let $O{E}_g$ (resp. $C{E}_g$ and $A{E}_g$) and resp. $O{E}_g^o$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces ${\mathrm{\Sigma }}_g$ which extend over $(S^3,{\mathrm{\Sigma }}_g)$ among all embeddings ${\mathrm{\Sigma }}_g\to S^3$ and resp. unknotted embeddings ${\mathrm{\Sigma }}_g\to S^3$. It is known that $O{E}_g^o\le 12(g-1)$, and we show that $12(g-1)$ is reached for an unknotted embedding ${\mathrm{\Sigma }}_g\to S^3$ if and only if $g=2$, 3, 4, 5, 6, 9, 11, 17, 25, 97, 121, 241, 601. Moreover $A{E}_g$ is $2g+2$; and $C{E}_g$ is $2g+2$ for even $g$, and $2g-2$ for odd $g$. Efforts are made to see intuitively how these maximal symmetries are embedded into the symmetries of the 3-sphere.