Real and Quasi-central Elements of Finite Groups

An element [Formula: see text] of a finite group [Formula: see text] is said to be real in [Formula: see text] if [Formula: see text] for some [Formula: see text] in [Formula: see text], and quasi-central if [Formula: see text] for each [Formula: see text] in [Formula: see text]. When [Formula: see text] is a 2-group, [Formula: see text] and [Formula: see text], we prove that if every real element in [Formula: see text] of order 4 is quasi-central in [Formula: see text], then every element [Form