### Abstract

Let k be an algebraically closed field of prime characteristic p . Let G be a finite group, let N be a normal subgroup of G , and let c be a G -stable block of kN so that (kN)c{(kN)c} is a p -permutation G -algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a (G,N,c){(G,N,c)}-Brauer pair (R,fR){(R,f_{R})} consists of a p -subgroup R of G and a block fR{f_{R}} of (kCN(R)){(kC_{N}(R))}. If Q is a defect group of c and fQ∈𝐵ℓ(kCN(Q)){f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))}, then (Q,fQ){(Q,f_{Q})} is a (G,N,c){(G,N,c)}-Brauer pair. The (G,N,c){(G,N,c)}-Brauer pairs form a (finite) poset. Set H=NG(Q,fQ){H=N_{G}(Q,f_{Q})} so that (Q,fQ){(Q,f_{Q})} is an (H,CN(Q),fQ){(H,C_{N}(Q),f_{Q})}-Brauer pair. We extend Lemma 8.6.4 of the above book to show that if (U,fU){(U,f_{U})} is a maximal (G,N,c){(G,N,c)}-Brauer pair containing (Q,fQ){(Q,f_{Q})}, then (U,fU){(U,f_{U})} is a maximal (H,CN(c),fQ){(H,C_{N}(c),f_{Q})}-Brauer pair containing (Q,fQ){(Q,f_{Q})} and conversely. Our main result shows that the subcategories of ℱ(U,fU)(G,N,c){\mathcal{F}_{(U,f_{U})}(G,N,c)} and ℱ(U,fU)(H,CN(Q),fQ){\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including (Q,fQ){(Q,f_{Q})} and (U,fU){(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.