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Let $\mathbb{C}\mathsf{A}_n = \mathbb{C}[S_2\wr S_2 \wr\cdots \wr S_2]$ bethe group algebra of $n$-step iterated wreath product. We prove some structuralproperties of $\mathsf{A}_n$ such as their centers, centralizers, right anddouble cosets. We apply these results to explicitly write down Mackey theoremfor groups $\mathsf{A}_n$ and give a partial description of the naturaltransformations between induction and restriction functors on therepresentations of the iterated wreath product tower by computing certain homspaces of the category of $\displaystyle \bigoplus_{m\geq 0}(\mathsf{A}_m,\mathsf{A}_n)-$bimodules. A complete description of the category is an openproblem.

Natural transformations between induction and restriction on iterated wreath product of symmetric group of order $2$