Abelian p-group with indecomposable coprime automorphism group is homocyclic

Statement
Suppose $$P$$ is an Abelian group of prime power order and $$A \le \operatorname{Aut}(P)$$ is a group of order relatively prime to $$p$$, such that $$A$$ acts indecomposably on $$P$$. In other words, $$P$$ cannot be expressed as an internal direct product of $$A$$-invariant subgroups. Then, $$P$$ is a homocyclic group.

Textbook references

 * , Page 176, Theorem 2.2, Section 5.2 ($$p'$$-automorphisms of Abelian $$p$$-groups)