Extending periodic automorphisms of surfaces to 3-manifolds

In the orientable category, we show that, if a finite group G acts on a closed surface Σ so that each element of it is extendable over the 3-sphere S³as a diffeomorphism with respect to a fixed embedding Σ → S³, then G is extendable over S³as a group action with respect to some other embedding Σ → S³. Degree one maps are involved in the proof.