Linking the residual finiteness of $G$ with $Aut(G)$ or $Out(G)$

Question

There is a classic result of Baumslag which states,

Thm: If $G$ is residually finite then so is $\operatorname{Aut}(G)$.

While Grossman proved the (essentially) analogous result for $\operatorname{Out}(G)$,

Thm: If $G$ is conjugacy separable and every conjugating automorphism is inner then $\operatorname{Out}(G)$ is residually finite.

(A conjugating automorphism is an automorphism $\delta: g\mapsto g^{w_g}$ where $w_g$ is dependent on the $g\in G$ - so every element is sent to a conjugate of itself).

I was wondering if it was possible to "go the other way", so to speak,

What conditions, if any, can we put on $\operatorname{Aut}(G)$ or $\operatorname{Out}(G)$ to ensure that $G$ is residually finite?

One obvious condition is that if $\operatorname{Aut}(G)$ is residually finite and $G$ is centerless then $G$ is residually finite. However, you have the added stipulation that $G$ is centerless, which is a condition on $G$ not on $\operatorname{Aut}(G)$. That said, this is a relatively harmless condition on $G$ (checking $Z(G)=1$ is often easier than checking $G$ is residually finite). So I suppose my question can be taken up to "relatively harmless conditions on $G$".

Answer 1

First, Baumslag's result is for finitely generated groups only. HW already says that essentially the Out of a residually finite group can be ``arbitrary". Now if you take a Tarski monster with trivial Out, then the direct product of it with the residually finite group above gives a non-residualy finite group with an arbitrary Out. Another way is to use Minasyan's result from Minasyan, Ashot, Groups with finitely many conjugacy classes and their automorphisms. Comment. Math. Helv. 84 (2009), no. 2, 259–296. He constructed a 2-generated group with two conjugacy classes (hence simple and not residually finite) with arbitrary countable Out. So there is no connection between properties of Out and residual finiteness of the group.

With Aut the situation is more complicated since if the group is not res. finite and does not have center then Aut is not residually finite. On the other hand, there are non-residually finite f.g.groups with residualy finite Aut. For example, Anna Erschler, I think, constructed such a group as a central extension of a Grigorchuk group. See Erschler, Anna Not residually finite groups of intermediate growth, commensurability and non-geometricity. J. Algebra 272 (2004), no. 1, 154–172.