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We show that for a finite group $G$, the commuting probability of $G$ can beexplicitly bounded from below in a nontrivial way by a function in the maximumfraction of elements inverted resp. squared by an automorphism of $G$. Usingthese bounds together with a result of Guralnick and Robinson gives upperbounds on the index of the Fitting subgroup of $G$ under each of the twoconditions that $G$ have an automorphism inverting resp. squaring at least$\rho|G|$ many elements in $G$, for $\rho\in\left(0,1\right]$ fixed. This is animprovement on previous results of the author.

A note on finite groups with an automorphism inverting or squaring a non-negligible fraction of elements