Commuting fraction equals five-eighths iff inner automorphism group is Klein four-group

Statement
The following are equivalent for a finite group $$G$$:


 * 1) The fact about::commuting fraction of $$G$$ is exactly equal to $$5/8$$.
 * 2) The fact about::inner automorphism group of $$G$$ has order equal to four.
 * 3) The fact about::inner automorphism group of $$G$$ is a Klein four-group.

The condition defines a family of isoclinic groups. The groups in this family that are 2-groups form the $$\Gamma_2$$ family in the Hall-Senior nomenclature.

Related facts

 * Commuting fraction more than five-eighths implies abelian