$2$-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle II

Abstract: We consider $2$-blocks of finite groups with defect group $D=Q \times R$ and inertial quotient $\mathbb{E}$ where $Q \cong (C_{2<sup>m})<sup>n$, $R \cong C_{2<sup>r}$, and $\mathbb{E}$ contains a Singer cycle of $\operatorname{Aut}(Q)$ (an element of order $2^n-1$). We classify such blocks up to Morita equivalence when either $\mathbb{E}$ is cyclic or $r=1$. We achieve a partial classification when $r&gt;1$ and $E$ is non-cyclic.