Summary: In this work we are concerned with the singular quasilinear elliptic equations of the type \[-\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=\frac{\mu+h(x)}{|x|^{(a+1)p}}|u|^{p-2}u+k(x)\frac{|u|^{q-2}u}{|x|^{bq}},\ x\in \mathbb{R}^N,\] where \(1<p<N,\ 0\le a<\frac{N-p}p,\ a\le b<a+1,\ 0\le \mu<\overline{\mu}=\left(\frac{N-p}p-a\right)^p,\ q=p^*(a,b)=\frac{Np}{N-(1+a-b)p}\), and \(h\) and \(k\) are continuously bounded functions satisfying some symmetry conditions with respect to a closed subgroup \(G\) of \(O(N)\). By variational methods and the Caffarelli-Kohn-Nirenberg inequality, we obtain several existence and multiplicity results of \(G\)-symmetric solutions under certain appropriate assumptions on \(h\) and \(k\).