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Let $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$ be the group ofautomorphisms on the first homology group with $\mathbb Z_2$ coefficient of aclosed non-orientable surface $N_g$ preserving the mod $2$ intersection form.In this paper, we obtain a finite presentation for$\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$. As applications we calculatethe first homology group and the second homology group of$\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$.

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