A version of Brauer's second main theorem is as follows:



> 
> Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
> 
> 
> If $d\_{\chi\mu}^x\neq 0$ and $\mu$ belongs to a block $b$ of $C\_G(x)$, then $b^G=B$. Hence we have
> $$\chi(xy)=\sum\limits\_{b^G=B} \sum\limits\_{\mu\in\text{IBr}(b)} d\_{\chi\mu}^x\cdot \mu(y)$$
> for all $y\in C\_G(x)\_{p'}$.
> 
> 
> 


Now, let $B$ be a nilpotent 2-block of $G$ having $D:=C\_2\times C\_2$ as defect group, $x$ an involution in $G$, $y:=1$ and $b^G=B$.


**Questions:**



> 
> i) What can be said about $b$?
> 
> 
> ii) Is $l(b)$ or $k(b)$ (or bounds for these numbers) known?
> 
> 
> iii) Does $b$ (under certain circumstances) have to be nilpotent as well?
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> 
> iv) I think that the defect group $\tilde{D}$ of $b$ is contained in $D$. Do we have $D\cong\tilde{D}$?
> 
> 
> 


I would be thankful for references in the literature.


Thanks for the help.



Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks
A Brauer correspondent of a nilpotent block is always nilpotent, so $b$ is indeed nilpotent and in this case we $\ell(b) = 1$, $k(b) = 4$. Also, $b$ does indeed have defect group $D$.


Probably the best reference is to fully understand the Brou'e Puig Inventiones(?) paper on nilpotent blocks in this special case, although Brauer's paper on blocks with Abelian defect group and inertial index one (which is essentially equivalent to being nilpotent in the Abelian defect group case), which is an earlier paper, already provides answers to your question in the Klein $4$ defect group case (in the case of question iii), you obtain that $b$ has Abelian defect group $D$ with inertial index one).


