For various values of $n$ and $m$ we investigate homomorphisms $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ and $\mathrm{Out}(F_n)\to \mathrm{GL}_m(K)$, i.e. the free and linear representations of $\mathrm{Out}(F_n)$ respectively.

By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of $\mathrm{Out}(F_n)$ we prove that each homomorphism $\mathrm{Out}(F_n) \to \mathrm{GL}_m(K)$ factors through the natural map $\pi_n \colon \mathrm{Out}(F_n)\to \mathrm{GL}(H_1(F_n,\mathbb Z))\cong \mathrm{GL}_n(\mathbb Z)$ whenever $n= 3,m <7$ and $\mathrm{char}(K)\not\in \{2,3\}$, and whenever $n >5,m <\binom {n+1} 2$ and $\mathrm{char}(K) \not\in \{2,3,...,n+ 1\}$.

We also construct a new infinite family of linear representations of $\mathrm{Out}(F_n)$ (where $n >2$), which do not factor through $\pi_n$. When $n$ is odd these have the smallest dimension among all known representations of $\mathrm{Out}(F_n)$ with this property.

Using the above results we establish that the image of every homomorphism $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ is finite whenever $n= 3$ and $n < m <6$, and of cardinality at most $2$ whenever $n >5$ and $n < m < \binom n 2$. We further show that the image is finite when $\binom n 2 \leqslant m < \binom{n+1} 2$.

We also consider the structure of normal finite index subgroups of $\mathrm{Out}(F_n)$. If $N$ is such then we prove that if the derived subgroup of the intersection of $N$ with the Torelli subgroup $\overline{\mathrm{IA}}_n < \mathrm{Out(F_n)}$ contains some term of the lower central series of $\overline{\mathrm{IA}}_n$ then the abelianisation of $N$ is finite.