Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi\_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ which are not in the inner set, given an outer automorphism $\phi(x)$, we can create a new group which has the presentation $(Gen \cup \{\psi\}|Rel \cup \{ \psi x \psi^{-1} = \phi(x) | x \in Gen\})$. I've been referring to this notion by saying we want to internalize the outer automorphism $\phi(x)$. Some questions I've had so far are as follows:


What properties of the original group are preserved by this process?


Do most groups become trivial or infinite? (So far its looking like the latter)


I find this to be really difficult so I really hope somebody here will have some answers for these questions, or at least a place to look or a reason to expect this to be very difficult to answer. So far I've done this for some cyclic groups and they become evidently massive, though I can't prove that because I don't know how, as I just started reading about finite presentations and this sort of thing recently.



What happens when you internalize outer automorphisms?
Your group is just the semidirect product $\Gamma=G\rtimes\_\psi\mathbb{Z}$. It's easy to check that the isomorphism class of $\Gamma$ only depends on the conjugacy class of $\psi$ in $\mathrm{Out}(G)$; in particular, if $\psi$ were inner this would be the direct product.


One trivial corollary is that such $\Gamma$ is always infinite. (You could also have seen this using presentations by spotting the homomorphism to $\mathbb{Z}$ that kills $G$.)


