(I will use additive notation. In particular, $n\mid x$ for $n\in\mathbb N$ and $x\in G$ means there exists $y$ such that $ny=x$.)

It follows from [Szmielew’s quantifier elimination](https://mathoverflow.net/a/402100) that two elements $x,y$ of an abelian group $G$ *satisfy the same first-order formulas* if and only if
$$m\mid nx\iff m\mid ny$$
for all integers $n,m\ge0$; this reduces to the following two special cases:
1. $x$ and $y$ have the same order (i.e., $nx=0\iff ny=0$ for all  $n$).
2. $p^l\mid p^kx\iff p^l\mid p^ky$ for all primes $p$ and $l>k\ge0$ such that $p^k$ divides the order of $x$ (or $y$).

Now, if $G$ is *finite*, this is equivalent to the existence of an automorphism $\phi$ such that $\phi(x)=y$, because [elementary equivalent finite structures are isomorphic](https://math.stackexchange.com/a/1518666). Moreover, in a finite group, condition 2 implies condition 1.

(I’m sure there is a purely algebraic proof of this using the classification of finite abelian groups and the like, but well, I’m a logician, not an algebraist.)