On Bousfield's problem for solvable groups of finite Prüfer rank

For a group G and R=Z,Z/p,Q we denote by GˆR the R-completion of G. We study the map Hn(G,K)→Hn(GˆR,K), where (R,K)=(Z,Z/p),(Z/p,Z/p),(Q,Q). We prove that H2(G,K)→H2(GˆR,K) is an epimorphism for a finitely generated solvable group G of finite Prüfer rank. In particular, Bousfield's HK-localisation of such groups coincides with the K-completion for K=Z/p,Q. Moreover, we prove that Hn(G,K)→Hn(GˆR,K) is an epimorphism for any n if G is a finitely presented group of the form G=M⋊C, where C is the infinite cyclic group and M is a C-module.