Non-abelian group generated by aperiodic elements has no nontrivial power automorphism

Statement
Suppose $$G$$ is a non-abelian group generated by a set all of whose elements are aperiodic, i.e., they have infinite order. Then, $$G$$ has no power automorphisms other than the identity map.

Facts used

 * 1) uses::Cooper's theorem: This states that every power automorphism of a group commutes with every inner automorphism.