Extending periodic maps on surfaces over the 4-sphere

The topic indicated by the title has been addressed by Montesinos (1982) and Hirose (2002) using Rohlin intersection form, and by Ding-Liu-Wang-Yao (2012) using spin structures.
Based on the work above, we deduce a more computable criterion for extending periodic maps on surfaces over the 4-sphere.
Some applications are given. Results including:
(1) Let $F_g$ be the closed orientable surface of genus $g$ and $w_g$ be a periodic map of maximum order $F_g$.
Then $w_g$ is extendable over $S^4$ for some smooth embedding $e: F_g \to S^4$ if and only if $g=4k, 4k+3$.
(2) For infinitely many primes, each periodic map of order $p$ on $F_g$ is extendable over $S^4$ for some smooth embedding $e: F_g \to S^4$.
This is joint work with Zhongzi Wang.