  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.

Combinatorics;Representation Theory