Let $G$ be a (possibly infinite) non-centerless group, i.e. such that $Z(G) \ne \lbrace e \rbrace$. Left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \rbrace \le \operatorname{Sym}(G)$ and $\Gamma:=\lbrace \gamma_a \mid a \in G \rbrace \le \operatorname{Sym}(G)$, such that:
- $G \cong \Theta$ and $G \cong \Gamma$;
- $\Theta\Gamma=\Gamma\Theta \le \operatorname{Sym}(G)$;
- $Z(G) \cong \Theta \cap \Gamma$;
- $\operatorname{Inn}(G)= \Theta\Gamma \cap \operatorname{Aut}(G)$.
Can we state anything about $Z(\operatorname{Aut}(G))$ and, in particular, about its relationships with $\Theta \cap \Gamma$ (inclusion, intersection, etc.)?