Let $G,H$ be infinite finitely generated groups such that $F(G)=F(H)$. Where $F(G)$ denotes the isomorphism classes of finite quotients of $G$. Let $G$ be residually finite. can we say that $H$ is residually finite too?

As Steve D observes, the answer is 'no'. This is a very general phenomenon. Let $G$ be any finitely generated residually finite group, and let $S$ be any finitely generated group with no finite quotients (Higman gave an example, or use an infinite simple group). Then $F(G)=F(G\times S)$, but $G\times S$ is certainly not residually finite. 


$\mathbb{Z}\times\mathbb{Z}$ and Thompson's group $F$ have the same finite quotients, because $F'$ is the minimal normal subgroup of $F$. 

