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. 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.