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Questions about the group of automorphisms of any mathematical object $X$ endowed with a given structure, i.e the group of all bijective maps from $X$ to itself preserving this structure, and hence helping study it further and understand it better.

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Outer automorphisms of free groups into bigger free groups

Check out Bogopolski O.V. and Puga D.V., On embeddings of Out(F_n), the outer automorphism group of the free group of rank n, into Out(F_m) for m>n, Algebra and Logic, v. 41, no. 2 (2002), 69-73. …
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