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Let      F ( G ) \mathcal {F}(G)       denote the set of isomorphism classes of finite homomorphic images of a group  G . We say that groups  G  and  H  have isomorphic finite quotients if      F ( G ) = F ( H ) \mathcal {F}(G) = \mathcal {F}(H)      . Let      H \mathcal {H}       denote the class of finite extensions of finitely generated nilpotent groups. In this paper we show that if  G  is in      H \mathcal {H}      , then the groups  H  in      H \mathcal {H}       for which      F ( G ) = F ( H ) \mathcal {F}(G) = \mathcal {F}(H)       lie in only finitely many isomorphism classes.

Nilpotent-by-finite groups with isomorphic finite quotients