No, that does not follow. It is true that G can be identified with the set , but the group operation can be different. In , the usual group operation is given by But you can also define a group operation by The homomorphism from to is given in both cases by

In general, if is a surjective homomorphism with kernel K, then G is (isomorphic to) a semidirect product of K and H. The above example is the semidirect product , where the action of on itself comes from the automorphism of the additive group