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Let $G$ be a finite group and let $Z^3(G,U(1))$ be the set (indeed, group under pointwise multiplication) of all normalized 3-cocycles $G\times G\times G\to U(1)$, with trivial action on $U(1)$. For any normal subgroup $N$ of $G$, the canonical projection $G\to G/N$ induces an injection $Z^3(G/N,U(1))\to Z^3(G,U(1))$ in the usual way. Note that I am not interested in cohomology classes here, but in all (normalized) 3-cocycles.

Can this map ever be a surjection for $N$ non-trivial and proper? If so, does the answer change any if we suppose further that $N$ is central?

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