  We study the groups of rational points of abelian varieties defined over a
finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or,
equivalently, whose isogeny classes are determined by squarefree characteristic
polynomials. When $\mathrm{End}(A)$ is locally Gorenstein, we show that the
group structure of $A(\mathbb{F}_q)$ is determined by $\mathrm{End}(A)$.
Moreover, we prove that the same conclusion is attained if $\mathrm{End}(A)$
has local Cohen-Macaulay type at most $ 2$, under the additional assumption
that $A$ is ordinary or $q$ is prime. The result in the Gorenstein case is used
to characterize squarefree cyclic isogeny classes in terms of conductor ideals.
Going in the opposite direction, we characterize squarefree isogeny classes of
abelian varieties with $N$ rational points in which every abelian group of
order $N$ is realized as a group of rational points. Finally, we study when an
abelian variety $A$ over $\mathbb{F}_q$ and its dual $A^\vee$ succeed or fail
to satisfy several interrelated properties, namely $A\cong A^\vee$,
$A(\mathbb{F}_q)\cong A^\vee(\mathbb{F}_q)$, and
$\mathrm{End}(A)=\mathrm{End}(A^\vee)$. In the process, we exhibit a sufficient
condition for $A\not\cong A^\vee$ involving the local Cohen-Macaulay type of
$\mathrm{End}(A)$. In particular, such an abelian variety $A$ is not a
Jacobian, or even principally polarizable.
