This thesis deals with the structure of intermediate C∗-sub-algebras B, either of the form Cλ∗(Γ)⊆B⊆A⋊rΓ or of the type C(Y)⋊rΓ⊆B⊆C(X)⋊rΓ. We begin by investigating the ideal structure of intermediate C∗-sub-algebras B of the form Cλ∗(Γ)⊆B⊆A⋊rΓ for commutative unital Γ-simple Γ-C∗-algebras A. In particular, we show that if Γ is a C∗-simple group, then every such intermediate C∗-sub-algebra B is simple. Continuing our perusal, we find examples of inclusions Cλ∗(Γ)⊆A⋊rΓ for which every intermediate C∗-sub-algebra B of the form Cλ∗(Γ)⊆B⊆A⋊rΓ is a crossed product. We show that for a large class of actions Γ↷A of C∗-simple groups Γ on unital C∗-algebras A, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate C∗-sub-algebra B, Cλ∗(Γ)⊆B⊆A⋊rΓ, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic C∗-simple group Γ on a von Neumann algebra M with separable predual, every intermediate vNa N, L(Γ)⊆N⊆M⋊Γ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate C∗-sub-algebras B of the form C(Y)⋊rΓ⊆B⊆C(X)⋊rΓ for an inclusion of unital Γ-simple Γ-C∗-algebras C(Y)⊂C(X). We introduce a notion of generalized Powers' averaging and show that it is equivalent to the simplicity of the crossed product C(X)⋊rΓ. As an application, we show that every intermediate C∗-sub-algebras B, C(Y)⋊rΓ⊆B⊆C(X)⋊rΓ is simple whenever C(Y)⋊rΓ is simple.