The following is simply a collection of results, which I haven’t checked just yet.
Let be a finite non-abelian group. Let be the number of ordered pairs such that , and be the total number of ordered pairs in . So the real number belongs to .
In 1979, Rusin has worked out explicitely the groups for which , in this freely available paper. The maximum value is , attained both by the quaternion group and the dihedral group . Then there is an infinite decreasing series starting at and accumulating at . And finally between and there are a handful of other cases.
Some problems regarding lower bounds, or the density of the values of in intervals with are mentionned, I believe they are still open.
The problem has been generalized more recently: to n-tuples in compact infinite groups by Erfanian and Russo, to pairs of subgroups by Tărnăuceanu, and to semigroups by Givens (in which case there is density in .)