  We consider $2$-blocks of finite groups with defect group $D=Q \times R$ and
inertial quotient $\mathbb{E}$ where $Q \cong (C_{2^m})^n$, $R \cong C_{2^r}$,
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 $r>1$
and $E$ is non-cyclic.
