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.