  We consider orientation-preserving actions of finite groups $G$ on pairs
$(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in
$S^3$. In a previous paper, we considered the case of closed, necessarily
orientable surfaces, determined for each genus $g>1$ the maximum order of such
a $G$ for all embeddings of a surface of genus $g$, and classified the
corresponding embeddings.
  In the present paper we obtain analogous results for the case of bordered
surfaces $\Sigma$ (i.e. with non-empty boundary, orientable or not). Now the
genus $g$ gets replaced by the algebraic genus $\alpha$ of $\Sigma$ (the rank
of its free fundamental group); for each $\alpha > 1$ we determine the maximum
order $m_\alpha$ of an action of $G$, classify the topological types of the
corresponding surfaces (topological genus, number of boundary components,
orientability) and their embeddings into $S^3$. For example, the maximal
possibility $12(\alpha - 1)$ is obtained for the finitely many values $\alpha =
2, 3, 4, 5, 9, 11, 25, 97, 121$ and $241$.
