  We continue and complete our previous paper `Lifts of projective congruence
groups' [2] concerning the question of whether there exist noncongruence
subgroups of $\SL_2(\Z)$ that are projectively equivalent to one of the groups
$\Gamma_0(N)$ or $\Gamma_1(N)$. A complete answer to this question is obtained:
In case of $\Gamma_0(N)$ such noncongruence subgroups exist precisely if
$N\not\in {3,4,8}$ 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 $\Gamma_1(N)$
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
$\SL_2(\Z)$ that can serve as basis for experimentation with modular forms on
noncongruence subgroups.
