Familiar Exercise: Suppose $G$ is a finite group and $T$ is an automorphism of $G$ which sends more than three quarters of elements of $G$ onto their inverses, then prove that $G$ is abelian.
The group of Quarternions is an example of a group which sends exactly $\displaystyle \frac{3}{4}$ elements onto their inverses.
Is there a finite group with an automorphism $T$ which sends exactly $\displaystyle\frac{4}{5}$ of elements of $G$ onto their inverses?
Similarly can we find groups with Automorphism $T$ which sends exactly $\displaystyle \frac{n}{n+1}$ of elements of $G$ onto their inverses.