Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 2 / b / Solution

The commutator subgroup $[G,G]$ is characteristic: every automorphism sends commutators to commutators and therefore preserves the subgroup they generate. Thus an automorphism $A$ inducesThis is independent of the coset representative $g$. If $A^{\prime }=c_h\circ A$ differs from $A$ by an inner automorphism, thenbecause the abelianization is abelian. Hence $q([A])=\overline{A}$ is independent of the representative of the outer class. Finally $\overline{AB}=\overline{A}\overline{B}$, sois a well-defined group homomorphism.