We study the action of the Artin braid group $B_{2g+2}$ on the set of spin structures on a hyperelliptic curve of genus $g$, which reduces to that of the symmetric group $S_{2g+2}$. It has been already described in terms of the classical theory of Riemann surfaces. In this paper, we compute the $S_{2g+2}$-orbits of the spin structures of genus $g$ and the isotropy group $\mathfrak{G}_i$ of each orbit in a purely combinatorial way.