Groups with the same cohomology as their pro-$p$ completions

For any prime $p$ and group $G$, denote the pro-$p$ completion of $G$ by $\hat{G}^p$. 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 $\mathcal{C}$. In particular, we show that right-angled Artin groups are in $\mathcal{C}$ and that this class also contains some special types of free products with amalgamation.