A note on involution centralizers in black box groups

Here we note a minor variation on the method in [J. N. Bray, An improved method for generating the centralizer of an involution, Arch. Math. (Basel) 74 2000, 4, 241–245] which enables calculations of CH(t) for H a subgroup of a black box group G and t an involution of G. We then give some applications of this, the first to investigate the action of the Lyons sporadic group via conjugation on its involutions, a number of calculations in E6(2) and E8(2), then finally to carry out some calculations in GL670(2).