I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question: name a group and an automorphism that takes $3/4$ of the elements of the group to their own inverses (for instance, the dihedral group $D_4$ of order $8$, with identity automorphism). I tried to figure out how to approach this question in general but can't see how.

1) Can we construct all such groups?

(it is asserted in comment to the answer below that these are precisely those finite groups whose center has index 4)

2) Given a rational number $a/b\in [0,1]$ does there exist a finite group $G$ and an automorphism $f$ such that $f$ maps exactly $a/b$ elements of $G$ to their own inverses?

($a/b=1$ is achieved precisely for the inversion map on an abelian group; otherwise $a/b\le 3/4$ according to the answer below)

3) Also, can these questions make sense in infinite groups?