Regular inclusions of simple unital C^*-algebras

We prove that an inclusion B ⊂ A of simple unital C^*-algebras with a finite-index 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 ⊂ Cʳα, 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 $2$.