Finite Autocapable Groups with the 2-Groups of Maximal Class

A group [Formula: see text] is said to be autocapable if there exists a group [Formula: see text] such that [Formula: see text] is isomorphic to the absolute central factor group [Formula: see text] of [Formula: see text]. In this paper, we first prove that if [Formula: see text] is a characteristic subgroup of an autocapable group [Formula: see text], then [Formula: see text] is neither the generalized quaternion group nor the semi-dihedral group. Next, we give the classification of finite groups [Formula: see text] if [Formula: see text] is a 2-group of maximal class.