Centralizer containment makes block induction well-defined

By F2, as an $H\times H$-module $M=kG$ splits over its double-coset orbits into $kH\oplus {⨁}_{t\mathrm{\notin }H}k[HtH]$. Each orbit module is induced from its point stabilizer: the map from stabilizer cosets to orbit points is a bijection, exactly as in F2. If an indecomposable summand of $k[HtH]$ had a vertex containing an $(H\times H)$-conjugate of $\mathrm{\Delta }D$, F3 would place that diagonal conjugate in a conjugate of the point stabilizer. Equivalently $\mathrm{\Delta }D$ would fix some point $t^{\mathrm{\prime }}\in HtH$, so $d{t}^{\mathrm{\prime }}=t^{\mathrm{\prime }}d$ for every $d\in D$. This puts $t^{\mathrm{\prime }}\in C_G(D)\le H$, impossible when $t\mathrm{\notin }H$. The same argument proves the exclusion for any $p$-subgroup $Q\le H$ with $C_G(Q)\le H$, replacing $D$ by $Q$.

The identity orbit $kH$ contains $b$ exactly once in its block decomposition, by F1. By F4, F9 and step 1.1 no other orbit contains an isomorphic summand. Thus $b$ has multiplicity one in $M$. Decomposing $M=kG$ into global blocks and applying F8 shows exactly one global block $B$ has $b$ as a restriction summand. By F1 this is $b^G$. This proves existence without the extra normalizer assumption.

We identify its central character carefully. Use the decomposition $M=b\oplus X$, where the projection onto $b$ is $r(a)=e{\pi }_H(a)$ and ${\pi }_H$ deletes coefficients outside $H$. Decompose $X$ into indecomposables $X_j$ by F8; none is isomorphic to $b$ by step 2.1. Any composite $b\to X_j\to b$ is a nonunit in ${\mathrm{End}}_{H\times H}(b)$: if invertible it would split $b$ from $X_j$, forcing an isomorphism by indecomposability. This endomorphism ring is $Z(b)$, as in F6's block-center construction; its nonunits form the nilpotent ideal $J_b$. Therefore for $T\in {\mathrm{End}}_{H\times H}(M)$ the function $\chi (T)={\lambda }_b(rT{\mathrm{\mid }}_b)$ is a unital algebra homomorphism: in the corner of a composite, all cross-composites through the $X_j$ vanish modulo $J_b$, leaving the product of the two corners.

Let $E_C$ be projection of $M$ onto a global block $C$. If $C\mathrm{\ne }B$, decompose its restriction into indecomposables, none isomorphic to $b$, and step 3.1 gives $\chi (E_C)=0$. Since the projections sum to the identity, $\chi (E_B)=1$. For $z\in Z(kG)$ let $L_z$ be multiplication by $z$. On $B$, multiplication by $z-{\lambda }_B(z)$ is nilpotent by F6. Hence $(L_z-{\lambda }_B(z))E_B$ is nilpotent on $M$, and applying the field-valued homomorphism $\chi$ gives $\chi (L_z)={\lambda }_B(z)$. But its corner on the explicit $b\subset kH$ is multiplication by $e{\pi }_H(z)$, because ${\pi }_H$ is an $H$-bimodule projection. Thus ${\lambda }_B(z)={\lambda }_b({\pi }_H(z))\mathrm{.}$ Centrality of $z$ makes ${\pi }_H(z)\in Z(kH)$.

Now assume $H\le N_G(D)$, so $D⊴H$. Let $S$ be a simple $b$-module, whose existence and scalar character are proved in F6. F7 says every $d\in D$ acts trivially on $S$. Expand ${\pi }_H(z)$ in group elements and partition $H\setminus C_G(D)$ into conjugation orbits of $D$. Each orbit has size a power of $p$ greater than one; its coefficients in $z$ are constant. All conjugate elements have the same operator on $S$, so its orbit sum acts as zero in characteristic $p$. The remaining terms are precisely ${\mathrm{Br}}_D(z)$ by F5. Since $H$ normalizes $D$, it preserves $C_G(D)$ and this projection is central in $kH$. Consequently the two central elements have the same scalar on $S$, giving ${\lambda }_b({\pi }_H(z))={\lambda }_b({\mathrm{Br}}_D(z))$. Combine with step 4.1 to prove the formula.

For $D=1$, the containment assumption forces $H=G$ and all projections in the formula are identity. For $H=G$ induction is already the identity by F1. Empty off-identity double-coset families and empty noncentralizing orbit families simply contribute zero in the above sums. Every decomposition and orbit calculation is finite; no AC is added. The extra normalizer assumption was used only in step 5.1, so the formula has not been asserted outside its stated domain.