Steenrod ∪_i-products on Bredon - Illman cohomology

Let G be a topological group acting on a space X. We construct a family of Steenrod's ∪_i-product [Ann. of Math. (2) 48 (1947) 290] on the Bredon-Illman cochain complex of X [Quart. J. Math. Oxford Ser. (2) 47 (1996) 199]. As corollaries, we get the existence of Steenrod squares on Bredon-Illman cohomology with appropriate coefficients as well as the triviality of the Gerstenhaber bracket induced by the braces at the cochain level [G. Mukherjee, N. Pandey, Homotopy G-algebra structure on Bredon-Illman cochain complex, Preprint].