Abstract: Let S_g be the closed orientable surface of genus g, G be a finite group acting on it, and M be an integer homology 3-sphere. We show that in the orientable category if each element of G is extendable over M with respect to a fixed embedding from S_g to M, then G is extendable over some M' which is 1-dominated by M. It has several variations and generalizations. For example, S_g can be replaced by a connected compact manifold or a 3-connected graph. We also classify all orientation-preserving periodic automorphisms of S_g that are extendable over the 3-sphere. The corresponding embedding of such an automorphism can always be an unknotted one.
This is a jonit work with Yi Ni and Shicheng Wang.