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Let $\\mathcal{C}$ be the class of all groups $G$ such that, for each natural number $n$ and prime number $p$, $H^n(\\hat{G^p},\\mathbb Z/p)\\cong H^n(G, \\mathbb Z/p)$, where $\\mathbb Z/p$ is viewed as a discrete, trivial $\\hat{G}^p$-module. In this article we identify certain kinds of groups that lie in $\\mathcal{C}$. 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