P. Hegarty et D. Machale, 2-GROUPS IN WHICH AN AUTOMORPHISM INVERTS PRECISELY HALF THE ELEMENTS, Bulletin of the London Mathematical Society, 30, 1998, pp. 129-135
We classify all finite 2-groups G for which the maximum proportion of
elements inverted by an automorphism of G is a half. These groups cons
titute 10 isoclinism families.