Embedding surfaces into $S^3$ with maximum symmetry

We restrict our discussion to the orientable category. For $g>1$, let $O{E}_g$ be the maximum order of a finite group $G$ acting on the closed surface ${\mathrm{\Sigma }}_g$ of genus $g$ which extends over $(S^3,{\mathrm{\Sigma }}_g)$, where the maximum is taken over all possible embeddings ${\mathrm{\Sigma }}_g\hookrightarrow S^3$. We will determine $O{E}_g$ for each $g$, indeed the action realizing $O{E}_g$. In particular, with 23 exceptions, $O{E}_g$ is $4(g+1)$ if $g\mathrm{\ne }{k}^2$ or $4(\sqrt{g}+1{)}^2$ if $g=k^2$, and moreover $O{E}_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.