The soluble radical and orbits of certain maps on finite groups

For each element $u$ in a finite group $G$ define a map $\theta_u\colon G\to G$by$\theta_u(g)=[g^{-u},g]$andset$\Theta_G(u)={g\in G\mid \theta_u^n(g)=g \hbox{ for some } n>0}$\mathrm{.}Then$\theta_u$ induces a permutation of ${\mathrm{\Theta }}_G(u)$; let ${\beta }_G(u)$ be the number of orbits apart from $\{1\}$. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group $G$ is bounded in terms of the values of ${\beta }_G(u)$ for $2$-elements $u$.