Consider the cocycle $\alpha_3(g_a,g_b, g_c) \in H^3(Z_2 \times Z_2,  \mathbb{R} /\mathbb{Z})$ in the 3rd cohomology group of $Z_2 \times Z_2$, where $g_a=(g_{a1},g_{a2})$ and $g_b=(g_{b1},g_{b2})$, $g_c=(g_{c1},g_{c2})$.

Let $\alpha_3(g_a,g_b)$ to be a cup product form:
$$\alpha_3(g_a,g_b, g_c)=(-1)^{g_{a1}g_{b2}g_{c2}}.
$$

>**question: How can we trivialize a $H^3(Z_2 \times Z_2,  \mathbb{R} /\mathbb{Z})$'s cocycle $\alpha_3$ into coboundary 
$$\alpha_3= \delta \beta_2$$ 
in a large group quaternion $H_8$ by finding the explicit 2-cochain $\beta_2$?** 

Where we can regard the group homomorphism $$H_8 \to Z_2 \times Z_2$$ and use the fact that $H_8/Z_2=Z_2 \times Z_2$, we can call $H_8=G$, and $Z_2=N$ normal subgroup, and $Z_2 \times Z_2=Q$ as the quotient group. 

p.s. The original post in ME received little attention so I decide to try MO.