  As a contribution of the programme of Goswami and Mandal (2014), we carry out
explicit computations of $\mathbb{Q}(\Gamma,S)$, the quantum isometry group of
the canonical spectral triple on $C_{r}^{*}(\Gamma)$ coming from the word
length function corresponding to a finite generating set S, for several
interesting examples of $\Gamma$ not covered by the previous work Goswami and
Mandal (2014). These include the braid group of 3 generators,
$\mathbb{Z}_4^{*n}$ etc. Moreover, we give an alternative description of the
quantum groups $H_s^{+}(n,0)$ and $K_n^{+}$ (studied in Banica and Skalski
(2012), Banica and Skalski (2013)) in terms of free wreath product. In the last
section we give several new examples of groups for which $\mathbb{Q}(\Gamma)$
turn out to be doubling of $C^*(\Gamma)$.
