Quotient sets in nonabelian groups

We show that for a finite, nonempty subset A of a group, the quotient set A^-1A: =\a₁^{-1a₂ a₁, a₂ A\} has size |A^-1A|53\, |A|, unless A is densely contained in a coset, or in a union of two cosets of a finite subgroup.