On the representation theory of wreath products of finite group and symmetric group

Let $G\wr{S_N}$ be the wreath product of a finite group $G$ and the symmetric group $S_N$. The aim of this paper is to prove the branching theorem for the increasing sequence of finite groups $G\wr{S_1}\subset G\wr{S_2}\subset\dots\subset G\wr{S_N}\subset\dots $ and the analog of Young's orthogonal form for this case, using the inductive approach, invented by A. Vershik and A. Okounkov for the case of symmetric group.