Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.

Brauer second main theorem states

If $\chi\in Irr(\mathbb{C}G)$ is an ordinary character in the $p$-block $B\in Bl(\mathbb{F}G)$, $u\in G_p$ a $p$-element, $\phi\in IBr_p(C_G(u))$ an irreducible Brauer character in the $p$-block $b\in Bl(\mathbb{F}C_G(u))$, then $d_{\phi,\chi}^u \neq 0 \implies b^G = B$

Nagao's theorem states

If $e\in Z(\mathbb{F}G)$ is an idempotent, $M \in \mathbb{F}G\mathsf{-mod}$ a module with $eM=M$ and $H$ a subgroup with $C_G(D) \leq H \leq N_G(D)$ a subgroup and $Br_D: Z(\mathbb{F}G) \to Z(\mathbb{F}H)$ the Brauer homomorphism, then the restriction of $M$ to $H$ decomposes as $Res_H^G(M) = Br_D(e)M \oplus M'$ for some $\mathbb{F}H$-module $M'$ whose indecomposable summands all are relatively projective w.r.t. to some $Q\leq H$ with $D\not\leq Q$.

I have seen in several places, for example in Benson's book, that Nagao's theorem is considered to be "the module theoretic version of Brauer's second main theorem". I struggle to see the similarity. I assume Nagao's theorem (at least) implies the 2nd main theorem, but I am unable to prove it right now. Perhaps I'm missing some crucial inside about the connection between the generalised decomposition numbers and the Brauer homomorphism.

Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.

Brauer second main theorem states

If $\chi\in Irr(\mathbb{C}G)$ is an ordinary character in the $p$-block $B\in Bl(\mathbb{F}G)$, $u\in G_p$ a $p$-element, and $\phi\in IBr_p(C_G(u))$ an irreducible Brauer character in the $p$-block $b\in Bl(\mathbb{F}C_G(u))$, then $d_{\phi,\chi}^u \neq 0 \implies b^G = B$

Nagao's theorem states

If $e\in Z(\mathbb{F}G)$ is an idempotent, $M \in \mathbb{F}G\mathsf{-mod}$ a module with $eM=M$, $H$ a subgroup with $C_G(D) \leq H \leq N_G(D)$, and $Br_D: Z(\mathbb{F}G) \to Z(\mathbb{F}H)$ the Brauer homomorphism, then the restriction of $M$ to $H$ decomposes as $Res_H^G(M) = Br_D(e)M \oplus M'$ withsome $\mathbb{F}H$-module $M'$ whose indecomposable summands all are relatively projective w.r.t. to some $Q\leq H$ with $D\not\leq Q$.

I have seen in several places, for example in Benson's book, that Nagao's theorem is considered to be "the module theoretic version of Brauer's second main theorem". I struggle to see the similarity. I assume Nagao's theorem (at least) implies the 2nd main theorem, but I am unable to prove it right now. Perhaps I'm missing some crucial insight about the connection between the generalised decomposition numbers and the Brauer homomorphism.