  In this paper we investigate actions of $\operatorname{SAut}(F_n)$, the
unique index $2$ subgroup of $\operatorname{Aut}(F_n)$, on small sets,
improving upon results by Baumeister--Kielak--Pierro for several small values
of $n$. Using a computational approach for $n \geqslant 5$, we show that every
action of $\operatorname{SAut}(F_n)$ on a set containing fewer than $18$
elements is trivial.
