Two questions in the Kourovka Notebook

G. Glauberman's $Z^*$-theorem [J. Algebra, 4, No. 3, 403–420 (1966)] and the theorem of Bender are two most important tools for local analysis in the theory of finite groups. The $Z^*$-theorem generalizes the known Burnside and Brauer–Suzuki theorems on finite groups with cyclic and quaternion Sylow $2$-subgroups. Whether these theorems are valid in a class of periodic groups is unknown. We prove that the $Z^*$-theorem is invalid in the class of all periodic groups. In particular, this gives negative answers to questions of A. V. Borovik and V. D. Mazurov [see Unsolved Problems in Group Theory, The Kourovka Notebook, Questions 11.13 and 17.71a].