Direct limit of iterated automorphism groups Let $G$ be a group, and define a sequence of groups by $G_0=G, G_{i+1}=Aut(G_i)$. Define a homomorphism $f_i:G_i\to G_{i+1}$ by sending each $g\in G_i$ to the automorphism defined by $\varphi_g(x)=g^{-1}xg$. Then take the direct limit of the system
$G_0\to^{f_0} G_1\to^{f_1}...$
in the category of groups.
These are my questions:


*

*Is the group defined by this construction well defined?

*If it is well defined, is it nontrivial/not interesting? (for example, it is always 0 or always $G$?)


Edit: These two questions provide some interesting information about what the groups $G_i$ could be. 1
2 . It looks like the sequence very often stabilizes.
 A: Yes, this is well defined, and can even be continued past the natural numbers, through the ordinals! And it is very interesting. See e.g. this paper of Joel David Hamkins, and this question at Mathoverflow.
Specifically, you've defined the group $G_\omega$. It has an automorphism group too - that's $G_{\omega+1}$. We can continue this process "transfinitely" - so, $G_\alpha$ is well-defined for every ordinal $\alpha$. The sequence $(G_\alpha: \alpha\in ON)$ is the automorphism tower of $G$.
As far as I know, the main question about automorphism towers is when they "terminate" - that is, when (if ever) we hit some $\alpha$ so that $G_\alpha\cong G_{\alpha+1}$ via the canonical map from $G_\alpha$ to $G_{\alpha+1}$. For instance, if $G$ is finite and centerless (that is, the only element of $G$ commuting with every other element of $G$), then the automorphism tower terminates after finitely many steps; this was proved by Wielandt. Hamkins' result in the linked paper above is that every automorphism tower terminates.

As far as I know, continuing research on automorphism problems has a highly set-theoretic flavor.
The automorphism tower problem - which is still open - asks about the values of $\tau_\kappa$, the least upper bound on how long automorphism towers of groups of cardinality $<\kappa$ take to terminate. Computing $\tau_\kappa$ - indeed, under any reasonable additional axioms! - is extremely hard, and I don't believe has been done. Understanding the map $\kappa\mapsto \tau_\kappa$ is a major open problem.
Another question we can ask is whether we can "change" the height of the automorphism tower of a group via set-theoretic shenanigans (that is, forcing). Hamkins has studied this, and I believe others (Fuchs?) have as well.
