7.31 (1980)

Let $A$ be a group of automorphisms of a finite non-abelian 2-group $G$ acting transitively on the set of involutions of $G$. Is $A$ necessarily soluble?

Such groups $G$ were divided into several classes in (F. Gross, J. Algebra, 40, no. 2 (1976), 348–353). For one of these classes a positive answer was given by E. G. Bryukhanova (Algebra and Logic, 20, no. 1 (1981), 1–12).