Admissible subgroups of full ergodic groups

Let G be a countable group of automorphisms of a Lebesgue space (X, m) and let [G] be the full group of G. For a pair of countable ergodic subgroups W₁ and H₂ of [G], the following problem is considered: when are the full subgroups [H₁] and [H₂] conjugate in the normalizer N[G] = {g ∈ Aut X : g[G]g^-1 = [G]} of [G]. A complete solution of the problem is given in the case when [G] is an approximately finite group of type II and [H] is admissible, in the sense that there exists an ergodic subgroup [H₀] of [G] and a countable subgroup Γ ⊂ N[H₀] consisting of automorphisms which are outer for [H₀], such that [H₀] ⊂ [H] ⊂ [G] and the full subgroup [H₀, Γ] generated by [H₀] and Γ coincides with [G].