Normalizing isomorphisms between Burnside rings

We prove that if two finite groups G and G′ have isomorphic Burnside rings, then there is a normalized isomorphism between these rings, that is, a ring isomorphism θ :B(G)→B(G′) such that θ(G/1)=G′/1. We use this to prove that if two finite groups have isomorphic Burnside rings, then there is a one-to-one correspondence between their families of soluble subgroups which preserves order and conjugacy class of subgroups.