On the representation theory of $G\sim S_n$

In the Vershik-Okounkov approach to the complex irreducible representations of $S_n$ and $G\sim S_n$ we parametrize the irreducible representations and their bases by spectral objects rather than combinatorial objects and then, at the end, give a bijection between the spectral and combinatorial objects. The fundamental ideas are similar in both cases but there are additional technicalities involved in the $G\sim S_n$ case. This was carried out by Pushkarev. The present work gives a fully detailed exposition of Pushkarev's theory. For the most part we follow the original but our definition of a Gelfand-Tsetlin subspace, based on a multiplicity free chain of subgroups, is slightly different and leads to a more natural development of the theory. We also work out in detail an example, the generalized Johnson scheme, from this viewpoint.