For a given $${\epsilon > 0}$$ , we show that there exist two finite index subgroups of $${PSL_2(\mathbb{Z})}$$ which are $$({1+\epsilon})$$ -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $${\epsilon > 0}$$ there are two finite regular covers of the Modular once punctured torus T 0 (or just the Modular torus) and a $${(1+\epsilon)}$$ -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S p ) of the punctured solenoid S p under the action of the corresponding Modular group (which is the mapping class group of S p [6], [7]) has the closure in T(S p ) strictly larger than the orbit and that the closure is necessarily uncountable.