An involutory automorphism of a double cover of A8 with central fixed involutions

We construct a finite perfect group \\(E\\) with centre \\(\\{1,z\\}\\) and \\(E/\\langle z\\rangle\\cong A_8\\), together with an automorphism \\(\\alpha\\) of order two satisfying \\[\\{g\\in C_E(\\alpha):g^2=1\\}=\\{1,z\\}.\\] The group \\(E\\) contains a subgroup isomorphic to \\(C_2\\times C_2\\), so its Sylow \\(2\\)-subgroups are not generalized quaternion. Since \\(F^*(E)=E\\), this gives a counterexample to the assertion in Kourovka Problem 7.15. The construction uses the vectors \\(5(e_i-e_j)\\) in the Clifford algebra of the quadratic form \\(\\sum_{i=1}^8 x_i^2\\) over \\(\\mathbb F_7\\). The kernel of the permutation action is determined by exterior contractions and successive deletion of coordinates.