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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 numberof distinct element orders in $G$. In this paper, we are primarily concernedwith 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 viewedas 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 afunction in $\mathfrak{q}(G)$ and $\operatorname{o}(\operatorname{Rad}(G))$. Wealso obtain a curious quantitative characterisation of the Fischer-GriessMonster group $\operatorname{M}$.

Automorphism orbits and element orders in finite groups: almost-solubility and the Monster