Regular inclusions of simple unital C*-algebras

We prove that an inclusion B ⊂ A of simple unital C*-algebras with a finiteindex conditional expectation is regular if and only if there exists a finite group G that admits a cocycle action (α, σ) on the intermediate C*-subalgebra C generated by B and its centralizer CA(B) such that B is outerly α-invariant and (B ⊂ A) ∼= (B ⊂ C ⋊r (α,σ) G). Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most 2.