Groups with the same cohomology as their profinite completions

For any positive integer $n$, ${\mathcal{A}}_n$ is the class of all groups $G$ such that, for $0\le i\le n$, $H^i(\widehat{G},A)\cong H^i(G,A)$ for every finite discrete $\widehat{G}$-module $A$. We describe certain types of free products with amalgam and HNN extensions that are in some of the classes ${\mathcal{A}}_n$. In addition, we investigate the residually finite groups in the class ${\mathcal{A}}_2$.