Embedding surfaces into $S^3$ with maximum symmetry

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