Let $G$ be a group. We can define
$F(G)=\{g\in G\,|\,o(\operatorname{Cl}(g))<\infty\},$
where $o(\operatorname{Cl}(g))$ is the order (cardinality) of the conjugacy class of $g$ in $G$. This set is a subgroup of $G$ because
1) $o(\operatorname{Cl}(1))=1<\infty;$
2) we have $g(xy)g^{-1}=(gxg^{-1})(gyg^{-1}),$ so $\operatorname{Cl}(xy)\subseteq\operatorname{Cl}(x)\operatorname{Cl}(y).$ Therefore, if $o(\operatorname{Cl}(x))<\infty\text{ and }o(\operatorname{Cl}(y))<\infty,$ then $o(\operatorname{Cl}(xy))<\infty.$
$F(G)$ is a characteristic subgroup of $G.$ Indeed, let $\alpha$ be an automorphism of $G$. Then
$\alpha(g)\alpha(x)(\alpha(g))^{-1}=\alpha(gxg^{-1})$
for all $g,x\in G.$ We have $o(\alpha(\operatorname{Cl}(x)))=o(\operatorname{Cl}(x))$ because $\alpha$ is a bijection and so, since $\{\alpha(g)\,|\,g\in G\}=G,$ we obtain
$o(\operatorname{Cl}(\alpha(g)))<\infty,$
and therefore
$\alpha(g)\in F(G).$
I would like to ask if there is a name for this characteristic subgroup. Also, since $F(G)$ is a piece of ad hoc notation, I'd be grateful if you could tell me what the common notation is.
Is the group $G/F(G)$ important? What is it called?
And finally, where can I read about it?