Quotient group maps to outer automorphism group of normal subgroup
defined as follows: for any element of , pick an element in that coset of . Conjugation by induces an automorphism of , i.e., an element of the automorphism group . Although the automorphism depends on the choice of in the coset of , the coset of in for that element is independent of .
More explicitly, there is a composite map:
Under this map, the image of the subgroup of lies inside . Thus, we get an induced map from the quotient group to the quotient group .