Abelian inner mappings and nilpotency class greater than two

Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups. Kepka and Niemenmaa showed that if the inner mapping group of a finite loop QQ is abelian, then QQ is centrally nilpotent. Bruck showed that if the loop QQ is centrally nilpotent of class at most two, then the inner mapping group is abelian. In the 1990s Kepka raised the following problem: Is every finite loop with abelian inner mapping group centrally nilpotent of class at most two? The answer is: no. We construct the multiplication group of a loop of order 27 with abelian inner mapping group such that the loop is centrally nilpotent of class greater than two.