  Let $\mathbb{C}\mathsf{A}_n = \mathbb{C}[S_2\wr S_2 \wr\cdots \wr S_2]$ be
the group algebra of $n$-step iterated wreath product. We prove some structural
properties of $\mathsf{A}_n$ such as their centers, centralizers, right and
double cosets. We apply these results to explicitly write down Mackey theorem
for groups $\mathsf{A}_n$ 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 $\displaystyle \bigoplus_{m\geq 0}(\mathsf{A}_m,
\mathsf{A}_n)-$bimodules. A complete description of the category is an open
problem.
