Relative Brauer homomorphism

For $Q⊴P\le G$ with $P$ a $p$-group, put $B_Q=k{C}_G(Q)$ and define ${\mathrm{Br}}_{P\mathrm{/}Q}:B_Q^P⟶k{C}_G(P),{\sum }_{g\in C_G(Q)}{a}_{g}g⟼{\sum }_{g\in C_G(P)}{a}_{g}g\mathrm{.}$ Normality makes $P$ act on $C_G(Q)$, and $Q$ acts trivially, so this is a $P\mathrm{/}Q$ action. The orbit calculation in Brauer homomorphism is multiplicative applies to its permutation basis and proves that this map is a unital surjective algebra homomorphism. In particular a product coefficient at a $P$-fixed basis element has cancelling non-singleton $P\mathrm{/}Q$ orbits; the surviving pairs are in $C_G(P)$. This specifies the domain needed for a relative Brauer map, extending the notation in Brauer homomorphism for a p subgroup. When $Q=P$ it is the identity on $k{C}_G(P)$.

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245. Local argument and conventions as displayed above.