Groups with an automorphism which inverts at least one-third of the elements

Let [Formula: see text] be a finite group and [Formula: see text] be the set of the elements [Formula: see text] of [Formula: see text] such that [Formula: see text] where [Formula: see text]. In this paper, we give strong restrictions on the structure of the quotients [Formula: see text] where [Formula: see text] is a finite group with an automorphism [Formula: see text] such that [Formula: see text] and [Formula: see text] is the largest normal [Formula: see text]-subgroup of [Formula: see text].