We continue and complete our previous paper ``Lifts of projective congruence groups'' concerning the question of whether there exist noncongruence subgroups of  that are projectively equivalent to one of the groups  or . A complete answer to this question is obtained: In case of  such noncongruence subgroups exist precisely if  and we additionally have either that  or that  is divisible by an odd prime congruent to  modulo . In case of  these noncongruence subgroups exist precisely if .

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  that can serve as the basis for experimentation with modular forms on noncongruence subgroups.