TITLE: residually finite groups with the same finite quotients
QUESTION [8 upvotes]: Let $G , H$ be two finitely generated residually finite groups such that $F(G)=F(H)$. Where $F(G)$ denotes the isomorphism classes of finite quotients of $G$. Can we say that $G\cong H$?

REPLY [5 votes]: There are infinitely many metabelian groups with the same finite quotients, see Pickel, P. F.
Metabelian groups with the same finite quotients. 
Bull. Austral. Math. Soc. 11 (1974), 115–120.
On the other hand, for many relatively free groups, including free metabelian groups, the genus (i.e. the number of groups with the same finite quotients) is finite, see Gupta; Noskov, G. A.
On the genus of certain metabelian groups. Algebra Colloq. 5 (1998), no. 1, 49–66. 
See also Grunewald, Fritz; Zalesskii, Pavel;
Genus for groups. J. Algebra 326 (2011), 130–168.