2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174107Let 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).17 pages, 1 figure, minor mistake in closed genus 2 case corrected, references updatedGeometric TopologyGroup Theory57M99; 20F38text