Natural Transformations between Induction and Restriction on Iterated Wreath Product of Symmetric Group of Order 2

Let CA(n) = C[S-2 (sic) S-2 (sic) . . . (sic) S-2] be the group algebra of an n-step iterated wreath product. We prove some structural properties of A(n) such as their centers, centralizers, and right and double cosets. We apply these results to explicitly write down the Mackey theorem for groups A(n) and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of circle plus(m >= 0)(A(m), A(n))-bimodules. A complete description of the category is an open problem.