If $f$ is an automorphism and $|\{a: f(a) = a^{-1}\}| = 3/4 |G|$ then $G$ has an abelian subgroup of index $2$
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.