We study quantum isometry groups, denoted by Q(Γ, S), of spectral triples on C * r (Γ) for a finitely generated discrete group Γ coming from the word-length metric with respect to a symmetric generating set S. We first prove a few general results about Q(Γ, S) including:• For a group Γ with polynomial growth property, the dual of Q(Γ, S) has polynomial growth property provided the action of Q(Γ, S) on C * r (Γ) has full spectrum. • Q(Γ, S) ∼ = QISO(Γ, d) for any discrete abelian group Γ, where d is a suitable metric on the dual compact abelian groupΓ.We then carry out explicit computations of Q(Γ, S) for several classes of examples including free and direct product of cyclic groups, Baumslag-Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form Z k 2 or Z l 4 for some k, l in the direct product decomposition into cyclic subgroups.