Finite 2-Groups with Automorphisms of Order 4

It is proved that if a locally finite or locally nilpotent 2-group $G$ admits an automorphism $\varphi$ of order 4 with finitely many fixed points $m$ then $G$ possesses a normal subgroup $H$ of $m$-bounded index such that the second derived subgroup of $H$ is contained in its center.