Abelian-extensible automorphism not implies power map
For both these proofs, we use the fact that the group of rational numbers is a divisible group, and hence, is injective as a -module. Since a direct sum of injective modules is injective, the group is also injective. We combine these ideas with fact (1).
Rational multiplication on the rational numbers
On the group of rational numbers, consider an automorphism give by multiplication by a non-integer rational number. Such an automorphism is extensible (by fact (1), all automorphisms are extensible), but it is not a power map.
Coordinate exchange on two copies of the rational numbers
On the group , consider the automorphism , the coordinate exchange automorphism. Such an automorphism is extensible (by fact (1), all automorphisms are extensible), but it is not a power map. In fact, it is not even in the group of automorphisms generated by power maps.