I like to ask a simple question: **how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.**

Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 2-cocycle $\omega_2^G$ is a complex $U(1)=\mathbb{R}/\mathbb{Z}$ function with the norm $|\omega_2^G|=1$ but with a $U(1)$ complex phase satisfying the cocycle condition. Here $g_a, g_b \in G$.

How can we trivialize the 2-cocycle $\omega_2(g_a, g_b)$ of $G$ into 2-coboundary if we lift $G$ into a larger group $J$, and given that we know the group homomorphism $r$:

$$J \overset{r}{\rightarrow} G,$$

so that

$$\omega_2^J(j_a, j_b)=\omega_2^G(r(j_a),r(j_b))=\omega_2^G(g_a, g_b)=\delta v_1(j_a, j_b)=\frac{v_1(j_a) v_1(j_b)}{v_1(j_a j_b)} \text{ is trivial in } H^3(J,\mathbb{R}/\mathbb{Z}).$$

Namely, $\omega_2^J(j_a, j_b) =\delta v_1(j_a, j_b)$ is a 2-coboundary in $J$. I like to take $G=\mathbb{Z}_2 \oplus \mathbb{Z}_2$, and consider the group element $g_a=(g_{a1}, g_{a2}) \in (\mathbb{Z}_2, \mathbb{Z}_2)=\mathbb{Z}_2 \oplus \mathbb{Z}_2 =G$, similarly, $g_b=(g_{b1},g_{b2}) \in G$.

Let me focus on the 2-cocycle

$$ \omega_2^G(g_a, g_b)=\exp[i \pi \cdot g_{a1} \cdot g_{b2} ]=(-1)^{g_{a1} \cdot g_{b2} } \in H^2(G,\mathbb{R}/\mathbb{Z}), $$ is in a cup product form of $g_a, g_b$.

**Question: What are examples and restrictions of $J$? Whether dihedral group $D_8$ and quaternion $Q_8$ work for $J$? What is the explicit form of 1-cochain $v_1(j)$?**