Lifts of projective congruence groups, II

We continue and complete our previous paper `Lifts of projective congruence groups’ [2] concerning the question of whether there exist noncongruence subgroups of ${\text{SL}}_2(\mathbb{Z})$ that are projectively equivalent to one of the groups ${\mathrm{\Gamma }}_0(N)$ or ${\mathrm{\Gamma }}_1(N)$. A complete answer to this question is obtained: In case of ${\mathrm{\Gamma }}_0(N)$ such noncongruence subgroups exist precisely if $N\in ̸3,4,8$ and we additionally have either that $4\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of ${\mathrm{\Gamma }}_1(N)$ these noncongruence subgroups exist precisely if $N>4$. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of ${\text{SL}}_2(\mathbb{Z})$ that can serve as basis for experimentation with modular forms on noncongruence subgroups.