Strong Morita equivalence for inclusions of $C^{\ast }$-algebras induced by twisted actions of a countable discrete group

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^{\ast }$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^{\ast }$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^{\ast }$-algebras. Also, we suppose that one of the inclusions is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^{\ast }$-algebra in the multiplier $C^{\ast }$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.