  Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and
$M$ be an integer homology 3-sphere. We show that if each element of $G$ is
extendable over $M$ with respect to a fixed embedding $\Sigma\rightarrow M$,
then $G$ is extendable over some $M'$ which is 1-dominated by $M$. From this
result, in the orientable category we classify all periodic automorphisms of
closed surfaces that are extendable over the 3-sphere. The corresponding
embedded surface of such an automorphism can always be a Heegaard surface.
