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 over(G, ̂)^p. Let C be the class of all groups G such that, for each natural number n and prime number p, Hⁿ (over(G^p, ̂), Z / p) ≅ Hⁿ (G, Z / p), where Z / p is viewed as a discrete, trivial over(G, ̂)^p-module. In this article we identify certain kinds of groups that lie in C. In particular, we show that right-angled Artin groups are in C and that this class also contains some special types of free products with amalgamation.