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Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$

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,\...
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