I would like a finite dimensional irreducible representation of ${PSL}_2(\mathbb{Z})$ which contains a proper sub-representation when restricted to the congruence subgroup $\Gamma(2)$, the kernel of $\mathrm{mod}_2:{PSL}_2(\mathbb{Z})\to {PSL}_2(\mathbb{Z}_2)$.

This would be used in constructing exact sequences of differential calculi on group algebras, and examining what relations exist between the differential invariants and the group structure. A seriously infinite noncommutative example would be really useful.

I have read the paper on the irreps of $B_3$ and ${PSL}_2(\mathbb{Z})$ by Imre Tuba and Hans Wenzl, but after calculations on some (I have not tried all of them!) of their examples, I cannot get a reducible rep for $\Gamma(2)$ which was not already reducible for ${PSL}_2(\mathbb{Z})$.