On the abelianizations of congruence subgroups of Aut(F2)

Let F n be the free group of rank n, and let Aut+(F n ) be its special automorphism group. For an epimorphism π : F n → G of the free group F n onto a finite group G we call \({\Gamma^+(G,\pi)=\{ \varphi \in {\rm Aut}^+(F_n) \mid \pi\varphi = \pi \}}\) the standard congruence subgroup of Aut+(F n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ+(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ+(G, π) ≤ Aut+(F 2) has infinite abelianization.