A finite abelian $p$-group $H$ is *homogenous* when it is the direct sum of cyclic groups of the same order $p^r$, i.e. $H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$. Every finite abelian $p$-group $G$ can be decomposed into homogenous factors
$$
G = H_{d_1} \oplus H_{d_2} \oplus \cdots \oplus H_{d_k}
$$
where $H_{d_i}$ has exponent $p^{d_i}$ and $d_1 > d_2 > \cdots > d_k > 0$. While this decomposition is unique up to isomorphism, unless $k = 1$ there are many distinct choices for the homogenous subgroups themselves.

Now suppose $\varphi$ is an automorphism of $G$ of order coprime to $p$. Must there be a homogenous decomposition that is invariant under $\varphi$?

If it helps, you can assume that the order of $\varphi$ is prime. For the intended application, all I really need is an invariant homogenous subgroup of the largest possible exponent.