On a composition of subfactors with group subfactors

Let M be a type II1 factor, G be a finite group, and N ⊆ M be an irreducible subfactor of finite index. We prove that the composed lattice of the intermediate subfactor lattice for the inclusion N ⊆ M and the subgroup lattice of G can be realized as an intermediate subfactor lattice of a certain composed subfactor of finite index, and this subfactor also has finite depth when N ⊆ M has finite depth.