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For a given $\epsilon >0$, we show that there exist two finite indexsubgroups of $PSL_2(\mathbb{Z})$ which are $(1+\epsilon)$-quasisymmetricallyconjugated and the conjugation homeomorphism is not conformal. This impliesthat for any $\epsilon>0$ there are two finite regular covers of the Modularonce punctured torus $T_0$ (or just the Modular torus) and a$(1+\epsilon)$-quasiconformal between them that is not homotopic to a conformalmap. As an application of the above results, we show that the orbit of thebasepoint in the Teichm\"uller space $T(\S)$ of the punctured solenoid $\S$under the action of the corresponding Modular group (which is the mapping classgroup of $\S$ \cite{NS}, \cite{Odd}) has the closure in $T(\S)$ strictly largerthan the orbit and that the closure is necessarily uncountable.

The Teichm\"uller distance between finite index subgroups of $PSL_2(\mathbb{Z})$