Inverse Ambiguous Functions and Automorphisms on Finite Groups

If G is a finite group, then a bijective function f : G → G is inverse ambiguous if and only if f(x)⁻¹ = f⁻¹(x) for all x ∈ G. We give a precise description when a finite group admits an inverse ambiguous function and when a finite group admits an inverse ambiguous automorphism.