  For a finite group $G$, we denote by $\omega(G)$ the number of
$\operatorname{Aut}(G)$-orbits on $G$, and by $\operatorname{o}(G)$ the number
of distinct element orders in $G$. In this paper, we are primarily concerned
with the two quantities $\mathfrak{d}(G):=\omega(G)-\operatorname{o}(G)$ and
$\mathfrak{q}(G):=\omega(G)/\operatorname{o}(G)$, each of which may be viewed
as a measure for how far $G$ is from being an AT-group in the sense of Zhang
(that is, a group with $\omega(G)=\operatorname{o}(G)$). We show that the index
$|G:\operatorname{Rad}(G)|$ of the soluble radical $\operatorname{Rad}(G)$ of
$G$ can be bounded from above both by a function in $\mathfrak{d}(G)$ and by a
function in $\mathfrak{q}(G)$ and $\operatorname{o}(\operatorname{Rad}(G))$. We
also obtain a curious quantitative characterisation of the Fischer-Griess
Monster group $\operatorname{M}$.
