Geodesic
Embedding the Outer Automorphism Group $\operatorname{Out}(F_n)$ of a Free Group of Rank $n$ in the Group $\operatorname{Out}(F_m)$ for $m>n$
Algebra i logika, Tome 41 (2002) no. 2, pp. 123-129

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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 $n-1$.
Keywords: group of outer automorphisms, free group. 
@article{AL_2002_41_2_a0,
     author = {O. V. Bogopolskii and D. V. Puga},
     title = {Embedding the {Outer} {Automorphism} {Group~}$\operatorname{Out}(F_n)$ of {a~Free} {Group} of {Rank~}$n$ in the {Group} $\operatorname{Out}(F_m)$ for $m>n$},
     journal = {Algebra i logika},
     pages = {123--129},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2002_41_2_a0/}
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O. V. Bogopolskii; D. V. Puga. Embedding the Outer Automorphism Group $\operatorname{Out}(F_n)$ of a Free Group of Rank $n$ in the Group $\operatorname{Out}(F_m)$ for $m>n$. Algebra i logika, Tome 41 (2002) no. 2, pp. 123-129. http://geodesic.mathdoc.fr/item/AL_2002_41_2_a0/