Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two generators, any product groups with one to be nonamenable, property (T) groups, inner amenable groups, wreath product groups like $\mathbb{Z}\wr F_2$, etc.
I want to know whether there are some examples on concrete computation of the second cohomology group $H^2(G, \mathbb{Z}G)$.
I only care the nontrivial case.
A related question is asked here.