If $\varphi$ is an automorphism of $G = \langle x_1, \ldots, x_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x_1, \ldots, x_n)$, $\overline{\varphi}$, with $$x_i\varphi=_G x_i\overline{\varphi}$$ for all $1\leq i\leq n$ then $\varphi$ is said to be a free automorphism on $x_1, \ldots, x_n$.

In Magnus, Karrass and Solitar's book Combinatorial Group Theory a whole 3 pages are devoted to the topic of "Free Automorphisms and Free Isomorphisms" (sec. 3.6). However, this section says little more than "they exist", and interestingly that every automorphism can be viewed as one, but not necessarily on the given set of generators (but necessarily on a set of generators no more than twice the size of the given set).

I was therefore wondering if there exists any further results in this area? Specifically if anyone has done any work on what the subgroup of free automorphisms of a given generating set must look like?"

If $\varphi$ is an automorphism of $G = \langle x_1, \ldots, x_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x_1, \ldots, x_n)$, $\overline{\varphi}$, with $$x_i\varphi=_G x_i\overline{\varphi}$$ for all $1\leq i\leq n$ then $\varphi$ is said to be a free automorphism on $x_1, \ldots, x_n$.

In Magnus, Karrass and Solitar's book Combinatorial Group Theory a whole 3 pages are devoted to the topic of "Free Automorphisms and Free Isomorphisms" (sec. 3.6). However, this section says little more than "they exist", and interestingly that every automorphism can be viewed as one, but not necessarily on the given set of generators (but necessarily on a set of generators no more than twice the size of the given set).

I was therefore wondering if there exists any further results about these automorphisms?