  We generalize the Cohen-Lenstra heuristics over function fields to \'{e}tale
group schemes $G$ (with the classical case of abelian groups corresponding to
constant group schemes). By using the results of
Ellenberg-Venkatesh-Westerland, we make progress towards the proof of these
heuristics. Moreover, by keeping track of the image of the Weil-pairing as an
element of $\wedge^2G(1)$, we formulate more refined heuristics which nicely
explain the deviation from the usual Cohen-Lenstra heuristics for abelian
$\ell$-groups in cases where $\ell\mid q-1$; the nature of this failure was
suggested already in the works of Garton, EVW, and others. On the purely large
random matrix side, we provide a natural model which has the correct moments,
and we conjecture that these moments uniquely determine a limiting probability
measure.
