Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 3-cocycle $\omega_3^G$ is a complex $U(1)=\mathbb{R}/\mathbb{Z}$ function with the norm $|\omega_3^G|=1$ but with a $U(1)$ complex phase satisfying the cocycle condition. Here $g_a, g_b, g_c \in G$.

We like to ask how can we trivialize the 3-cocycle $\omega_3(g_a, g_b, g_c)$ of $G$ into 3-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

$$\omega_3^J(j_a, j_b, j_c)=\omega_3^G(r(j_a),r(j_b),r(j_c))=\omega_3^G(g_a, g_b, g_c) \text{ is trivial in } H^3(J,\mathbb{R}/\mathbb{Z}).$$

I like to take $G=\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$, and consider the group element $g_a=(g_{a1}, g_{a2}, g_{a3}) \in (\mathbb{Z}_2, \mathbb{Z}_2, \mathbb{Z}_2)=\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 =G$, similarly, $g_b=(g_{b1},g_{b2},g_{b3}) \in G$ and $g_{c}=(g_{c1},g_{c2},g_{c3}) \in G$. Let me focus on the 3-cocycle

$$ \omega_3^G(g_a, g_b, g_c)=\exp[i \pi \cdot g_{a1} \cdot g_{b2} \cdot g_{c3}]=(-1)^{g_{a1} \cdot g_{b2} \cdot g_{c3}} \in H^3(G,\mathbb{R}/\mathbb{Z}), $$ is in a cup product form of $g_a, g_b, g_c$.

Here $\cdot$ is a usual product of multiplication. And $g_{a1} \cdot g_{b2} \cdot g_{c3} \in \{0,1\}= \mathbb{Z}_2$.

We wish to find that $\omega_3^G(r(j_a),r(j_b),r(j_c))$ becomes a 3-coboundary in $H^3(J,\mathbb{R}/\mathbb{Z})$ for the cohomology group of $J$, but $\omega_3^G(g_a,g_b,g_c)$ originally was not a 3-coboundary but was a 3-cocycle for the cohomology group of $G$. We hope to explicitly write $$ \omega_3^G(g_a,g_b,g_c)=\omega_3^G(r(j_a),r(j_b),r(j_c))= \frac{\beta_2^J(j_b,\; j_c)\beta_2^J(j_a,\; j_b j_c)}{\beta_2^J(j_a j_b,\; j_c) \beta_2^J(j_a,\; j_b)}. $$ Here $\beta_2^J(j_a,\; j_b)$ is a 2-cochain for $j_a, j_b, j_c \in J$, and that $g_a=r(j_a)$, $g_b=r(j_b)$, $g_c=r(j_c) \in G$.

Question:For example, can we find explicitly what is the minimal $J$, so we can trivialize a cup-product 3-cocycle $(-1)^{g_{a1} \cdot g_{b2} \cdot g_{c3}}$ of $G=\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ in a larger group $J$? Here $G$ is order 8 group. Perhaps $J$ is a group of order 16 at least?