The Teichm\"uller distance between finite index subgroups of $PS{L}_2(\mathbb{Z})$

For a given $ϵ>0$, we show that there exist two finite index subgroups of $PS{L}_2(\mathbb{Z})$ which are $(1+ϵ)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ϵ>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ϵ)$-quasiconformal 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 Teichm"uller space $T(\mathrm{\S })$ of the punctured solenoid $\mathrm{\S }$ under the action of the corresponding Modular group (which is the mapping class group of $\mathrm{\S }$ \cite{NS}, \cite{Odd}) has the closure in $T(\mathrm{\S })$ strictly larger than the orbit and that the closure is necessarily uncountable.