Centralizers of maximal order and nilpotent normal subgroups

We give a negative answer to Kourovka Problem 14.67. We construct a finite group G of order $660602880$, an involution a∈ G, and a nilpotent normal subgroup H of order $16384$ such that |CG(a)|=41287680≥ |CG(g)| for every g≠1, but H CG(a). The group H is a bilinear central extension of two copies of F₂⁴ by ²F₂⁴. The ambient group is obtained by adjoining the natural action of GL₄(2) and an involution interchanging the two copies. The centralizer comparison follows from fixed-space bounds on the exterior square and elementary orbit calculations.