  We compute a family of coverings with four ramification points, defined over
$\mathbb{Q}$, with regular Galois group $PSL_5(2)$.
  On the one hand, this is (to my knowledge) the first explicit polynomial with
group $PSL_5(2)$ over $\mathbb{Q}(t)$. On the other hand, it also positively
answers the question whether $PSL_5(2)$ is the monodromy group of a rational
function over $\mathbb{Q}$. At least this does not follow from considering
class triples in $PSL_5(2)$, as there are no rigid, rational genus-zero
triples. Also, for 4-tuples, our family is the only one with a Hurwitz curve of
genus zero (however it does not seem immediately clear without explicit
computations whether this curve can be defined as a rational curve over
$\mathbb{Q}$). There are also genus zero families with five branch points, and
maybe their Hurwitz spaces can be shown to have rational points; however, so
far I have not seen such arguments.
