When will automorphism group that fixes group element contain only identity element.

Question

$\newcommand{\Aut}{\mathrm{Aut}}$Let $C_{\Aut(G)}(x)=\{\alpha \in \Aut(G) \mid \alpha(x)=x\}$, where $\Aut(G)$ is automorphism group of $G$.

So, my question is for a non-abelian group is there exist any $x$ such that $C_{\Aut(G)}(x)= \{I\}$ or there does not exist any $x$ such that $C_{\Aut(G)}(x)= \{I\}$

Answer 1

Hint 1. $x$ centralises itself.

Hint 2. if $x\not\in Z(G)$ then there exists a map from $C_G(x)$ to $C_{\operatorname{Aut}(G)}(x)$ whose image is non-trivial (why?).