We consider finite group actions on the 3-sphere which leave invariant an embedded, compact, bounded surface of algebraic genus g > 1 (orientable or non-orientable), and determine for each g the maximum order of such an action. For example, the maximal possibility 12(g-1) is obtained for the finitely many values g = 2, 3, 4, 5, 9, 11, 25, 97, 121 and 241. For each g > 1, we classify the topological types of the surfaces and their embeddings into the 3-sphere.
