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

Let CAn=C[S2?S2???S2] be the group algebra of an n-step iterated wreath product. We prove some structural properties An such as their centers, centralizers, and right double cosets. apply these results to explicitly write down Mackey theorem for groups give a partial description natural transformations between induction restriction functors on representations product tower by computing certain hom spaces category ?m?0(Am,An)?bimodules. A complete is open problem.