  Let Mod(S) be the extended mapping class group of a surface S. For S the
twice-punctured torus, we show that there exists an isomorphism of finite index
subgroups of Mod(S) which is not the restriction of an inner automorphism. For
S a torus with at least three punctures, we show that every injection of a
finite index subgroup of Mod(S) into Mod(S) is the restriction of an inner
automorphism; this completes a program begun by Irmak. For all of the above
surfaces, we establish the co-Hopf property for finite index subgroups of
Mod(S).
