Let $\Sigma_{g}$ be an oriented connected closed surface
  of genus $g$ and $\mathcal {M}_{g}$ be the mapping class group
  of $\Sigma_{g}$; i.e.,the group of all isotopy classes of
  orientation-preserving self-diffeomorphisms of $\Sigma_{g}$.
  We define $SP_{g}[q]$  as the subgroup of $\mathcal {M}_{g}$
  consisting of mapping classes which preserve the given spin
  structure associated to the  quadratic form $q$ on
  $H_{1}(\Sigma_{g} ; \mathbb{Z}_{2})$. \\
  As is well-known, the automorphisms over
  $H_{1}(\Sigma_{g} ; \mathbb{Z}_{2})$
  form the $\mathbb{Z}_{2}$-symplectic group
  $Sp(2g; \mathbb{Z}_{2})$.  In this talk, we will observe
  automorphisms over  $H_{1}(\Sigma_{g} ; \mathbb{Z}_{2})$
  induced by the elements of $SP_{g}[q]$,
  which form the subgroup of $Sp(2g; \mathbb{Z}_{2})$.
  We will call this group
  the spin-preserving symplectic group, and
  determine it explicitly where the case is $g=1$ and $g=2$.
  We call the images of surfaces embedded in the $4$-shpere
  surface-knots. Lastly we give an application to the
  surface-knot of genus two.