Automorphism group of finitely generated free group

Definition
Let $$n$$ be a natural number. The automorphism group of the free group on $$n$$ generators is defined as the automorphism group of $$F_n$$, the free group on $$n$$ generators. A group is termed the automorphism group of a finitely generated free group if it is isomorphic to $$\operatorname{Aut}(F_n)$$ for some natural number $$n$$.

General relation with reduced free groups
Let $$\mathcal{V}$$ be any subvariety of the variety of groups. If $$F_n(\mathcal{V})$$ denotes the free algebra on $$n$$ generators in $$\mathcal{V}$$, then $$F_n(\mathcal{V})$$ is the quotient of $$F_n$$ by a verbal subgroup. Groups of the form $$F_n(\mathcal{V})$$ are termed reduced free groups.

There is a natural homomorphism from $$\operatorname{Aut}(F_n)$$ to $$\operatorname{Aut}(F_n(\mathcal{V}))$$, which sends an automorphism of the free group to the induced automorphism of $$F_n(\mathcal{V})$$. One way of seeing this is to observe that $$F_n(\mathcal{V})$$ is the quotient of $$F_n$$ by a verbal subgroup, which is in particular a characteristic subgroup, hence any automorphism of $$F_n$$ descends to an automorphism of the quotient.

However, this homomorphism is not necessarily surjective. For instance, if $$\mathcal{V}$$ is given as the variety of abelian groups where every element has order $$p$$ for some prime $$p \ge 5$$, there are automorphisms of $$F_1(\mathcal{V} = \mathbb{Z}/p\mathbb{Z}$$ that do not arise from automorphisms of $$F_1 = \mathbb{Z}$$.

Relation with free abelian groups
Free abelian groups are free algebras in the variety of abelian groups. The free abelian group of rank $$n$$ is isomorphic to $$\mathbb{Z}^n$$, and its automorphism group is isomorphic to $$GL(n,\mathbb{Z})$$. Thus, we have a homomorphism:

$$\operatorname{Aut}(F_n) \to GL(n,\mathbb{Z})$$.

It turns out that this automorphism is surjective -- in other words, every automorphism of the free abelian group on $$n$$ generators arises from an automorphism of the free group on $$n$$ generators.

IAPS structure
The collection of groups $$\operatorname{Aut}(F_n)$$ form an IAPS of groups. In other words, we can construct injective homomorphisms:

$$\Phi_{m,n}: \operatorname{Aut}(F_m) \times \operatorname{Aut}(F_n) \to \operatorname{Aut}(F_{m+n})$$

satisfying the associativity condition.