Cycle lengths in finite groups and the size of the solvable radical

We prove the following: For any $\rho\in\left(0,1\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $\rho\cdot|G|$, then the index of the solvable radical $\operatorname{Rad}(G)$ in $G$ is bounded from above in terms of $\rho$, and such a condition is strong enough to imply solvability of $G$ if and only if $\rho>\frac{1}{10}$. Furthermore, considering, for exponents $e\in\left(0,1\right)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G|^e$, such a condition is strong enough to imply $|\operatorname{Rad}(G)|\to\infty$ for $|G|\to\infty$ if and only if $e>\frac{1}{3}$. We also prove similar results for a larger class of bijective self-transformations of finite groups, so-called periodic affine maps.