There is a bunch of negative results here:

1. Khramtsov has shown that there are no embeddings for $m = n+1$ (when $n>1$)

2. Bridson and Vogtmann have shown that any homomorphism $Out(F_n) \to Out(F_m)$ factors through $\mathbb{Z} / 2 \mathbb{Z}$ if

a) $ m < n, n>2$

b) $n > 8, n< m < 2n-2$ for $n$ odd, and $n < m< 2n$ for $n $ even. This is a slightly stronger version of a result of Potapchik-Rapinchuk.

3. I have a result showing that in fact all such homomorphisms factor through $\mathbb{Z} / 2 \mathbb{Z}$ if $n > 5$ and $ n < m < {n \choose 2}$. I can also show that in the range ${n \choose 2} \leqslant m < {n \choose 2} + n$, all such homomorphisms have finite image.

There are two positive results as well, one by Aramayona-Leininger-Souto and one by Bogopol'skii-Puga and Bridson-Vogtmann; in both cases $m$ grows at least exponentially with $n$.

As far as I am aware, this is all that is known. I'd be happy to give you further details if you are interested.

There is a bunch of negative results here:

1. Khramtsov has shown that there are no embeddings for $m = n+1$ (when $n>1$)

2. Bridson and Vogtmann have shown that any homomorphism $Out(F_n) \to Out(F_m)$ factors through $\mathbb{Z} / 2 \mathbb{Z}$ if

a) $ m < n, n>2$

b) $n > 8, n< m < 2n-2$ for $n$ odd, and $n < m< 2n$ for $n $ even. This is a slightly stronger version of a result of Potapchik--Rapinchuk.

3. I have a result showing that in fact all such homomorphisms factor through $\mathbb{Z} / 2 \mathbb{Z}$ if $n > 5$ and $ n < m < {n \choose 2}$.

There are two positive results as well, one by Aramayona--Leininger--Souto and one by Bogopol'skii--Puga and Bridson--Vogtmann; in both cases $m$ grows at least exponentially with $n$.

As far as I am aware, this is all that is known. I'd be happy to give you further details if you are interested.

There is a bunch of negative results here:

1. Khramtsov have shown that there are no embeddings for $m = n+1$ (when $n>1$)

2. Bridson and Vogtmann have shown that any homomorphism $Out(F_n) \to Out(F_m)$ factors through $\mathbb{Z} / 2 \mathbb{Z}$ if

a) $ m < n, n>2$

b) $n > 8, n< m < 2n-2$ for $n$ odd, and $n < m< 2n$ for $n $ even. This is a slightly stronger version of a result of Potapchik-Rapinchuk.

3. I have a result showing that in fact all such homomorphisms factor through $\mathbb{Z} / 2 \mathbb{Z}$ if $n > 5$ and $ n < m < {n \choose 2}$. I can also show that in the range ${n \choose 2} \leqslant m < {n \choose 2} + n$, all such homomorphisms have finite image.

There are two positive results as well, one by Aramayona-Leininger-Souto and one by Bogopol'skii-Puga and Bridson-Vogtmann; in both cases $m$ grows at least exponentially with $n$.

As far as I am aware, this is all that is known. I'd be happy to give you further details if you are interested.

There is a bunch of negative results here:

1. Khramtsov has shown that there are no embeddings for $m = n+1$ (when $n>1$)

2. Bridson and Vogtmann have shown that any homomorphism $Out(F_n) \to Out(F_m)$ factors through $\mathbb{Z} / 2 \mathbb{Z}$ if

a) $ m < n, n>2$

b) $n > 8, n< m < 2n-2$ for $n$ odd, and $n < m< 2n$ for $n $ even. This is a slightly stronger version of a result of Potapchik-Rapinchuk.

3. I have a result showing that in fact all such homomorphisms factor through $\mathbb{Z} / 2 \mathbb{Z}$ if $n > 5$ and $ n < m < {n \choose 2}$. I can also show that in the range ${n \choose 2} \leqslant m < {n \choose 2} + n$, all such homomorphisms have finite image.

There are two positive results as well, one by Aramayona-Leininger-Souto and one by Bogopol'skii-Puga and Bridson-Vogtmann; in both cases $m$ grows at least exponentially with $n$.

As far as I am aware, this is all that is known. I'd be happy to give you further details if you are interested.