  We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is
$\mathrm{PSL}_n(\mathbb{Z}/2\mathbb{Z}) = \mathrm{L}_n(2)$, thus confirming a
conjecture of Mecchia--Zimmermann. In the course of the proof we give an
exponential (in $n$) lower bound for the cardinality of a set on which
$\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can
act non-trivially. We also offer new results on the representation theory of
$\mathrm{SAut(F_n)}$ in small dimensions over small, positive characteristics,
and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of Lie type
and algebraic groups in characteristic $2$.
