Inducing isomorphic permutation representations from non-equivalent G-sets

Given a finite group $G$, one may consider two non-equivalent finite $G$-sets and wonder if they can produce isomorphic complex permutation representations. This problem is equivalent to studying the kernel of a ring homomorphism from the Burnside ring to the representation ring. Elements in this kernel are called Brauer relations, and a collection of past literature manages to classify Brauer relations for all finite groups. This paper aims to investigate the kernel through a commutative diagram involving four ring homomorphisms and describe Brauer relations for Dihedral groups of order $2^n\ge 8$.