Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\omega_d^G$ is a complex $U(1)=\mathbb{R}/\mathbb{Z}$ function with the norm $|\omega_d^G|=1$ but with a $U(1)$ complex phase satisfying the cocycle condition $ \delta\omega_d^G=1$.

question: We like to ask

whether there always exists some Abelian group $N$ as a normal subgroupof some bigger group $J$, such that $G$ is the quotient group $$ \frac{J}{N}=G $$ andsuch that we can always trivialize (or split) the $d$-cocycle $\omega_d$ of $G$ into $d$-coboundary if we lift $G$ into a larger group $J$? Given that we know the group homomorphism $r$: $$J \overset{r}{\rightarrow} G.$$ Namely, $$ \omega_d^G(g_i,\dots)=\omega_d^G(r(j_i),\dots)= \delta \beta_{d-1}^J(j_i,\dots). $$ with $g \in G, j \in J$.

You are welcome to comment or answer the partial case, for example, when $d=2$, $d=3$ or $d=4$, and when $G$ is a finite group. Partial comments or answers are still welcome!