A finite presentation for the automorphism group of the first homology of a non-orientable surface over ${\mathbb{Z}}_2$ preserving the mod $2$ intersection form

Let $\mathrm{Aut}(H_1(N_g;{\mathbb{Z}}_2),\cdot )$ be the group of automorphisms on the first homology group with ${\mathbb{Z}}_2$ coefficient of a closed non-orientable surface $N_g$ preserving the mod $2$ intersection form. In this paper, we obtain a finite presentation for $\mathrm{Aut}(H_1(N_g;{\mathbb{Z}}_2),\cdot )$. As applications we calculate the first homology group and the second homology group of $\mathrm{Aut}(H_1(N_g;{\mathbb{Z}}_2),\cdot )$.