Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements

There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or cubed by a single automorphism) is large enough must be close to being abelian. In this paper, we show the following: Fix a real number $\rho$ with $0<\rho\leq 1$. Then a finite group $G$ with an automorphism inverting or squaring at least $\rho|G|$ of the elements in $G$ is "almost abelian" in the sense that both the index and the derived length of the solvable radical of $G$ are bounded. Furthermore, if $G$ has an automorphism cubing at least $\rho|G|$ of the elements in $G$, then $G$ is "almost solvable" in the sense that the index of the solvable radical of $G$ is bounded.