Genuine non-congruence subgroups of Drinfeld modular groups

Let A be the ring of elements in an algebraic function field K over a finite   field FqFq which are integral outside a fixed place ∞. In an earlier paper we have shown that the Drinfeld modular group  G=GL2(A)G=GL2(A) has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of G  . In addition, for all but finitely many cases we evaluate ngncs(G)ngncs(G), the smallest index of a normal genuine non-congruence subgroup of G, and compare it to the minimal index of an arbitrary normal non-congruence subgroup.