Embedding the Outer Automorphism Group $\operatorname{Out}(F_n)$ of a Free Group of Rank $n$ in the Group $\operatorname{Out}(F_m)$ for $m>n$

It is proved that for every $n\geqslant1$, the group $\operatorname{Out}(F_n)$ is embedded in the group $\operatorname{Out}(F_m)$ with $m=1+(n-1)k^n$, where $k$ is an arbitrary natural number coprime to $n-1$.