Abelian-extensible automorphism not implies power map

Statement
We can have an abelian group $$G$$ and an abelian-extensible automorphism $$\sigma$$ of $$G$$ such that $$\sigma$$ is not a power map.

Facts used

 * 1) uses::Injective module implies every automorphism is infinity-extensible

Proof
For both these proofs, we use the fact that the group $$\mathbb{Q}$$ of rational numbers is a divisible group, and hence, is injective as a $$\mathbb{Z}$$-module. Since a direct sum of injective modules is injective, the group $$\mathbb{Q} \oplus \mathbb{Q}$$ 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 $$\mathbb{Q} \oplus \mathbb{Q}$$, consider the automorphism $$(x,y) \mapsto (y,x)$$, 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.