  We study homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$, and
$\mathrm{GL}(m,K)$ for $m < 7$, where $K$ is a field of characteristic other
than 2 or 3. We conclude that all $K$-linear representations of dimension at
most 6 of $\mathrm{Out}(F_3)$ factor through $\mathrm{GL}(3,Z)$, and that all
homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$ have finite
image.
