Donovan's conjecture and extensions by the centralizer of a defect group

We consider Donovan’s conjecture in the context of blocks of groups $G$ with defect group $D$ and normal subgroups $N⊲G$ such that $G=C_D(D\cap N)N$, extending similar results for blocks with abelian defect groups. As an application we show that Donovan’s conjecture holds for blocks with defect groups of the form $Q_8\times C_{2^n}$ or $Q_8\times Q_8$ defined over a discrete valuation ring.