  We study a class of strictly weakly integral fusion categories
$\mathfrak{I}_{N, \zeta}$, where $N \geq 1$ is a natural number and $\zeta$ is
a $2^N$th root of unity, that we call $N$-Ising fusion categories. An $N$-Ising
fusion category has Frobenius-Perron dimension $2^{N+1}$ and is a graded
extension of a pointed fusion category of rank 2 by the cyclic group of order
$\mathbb Z_{2^N}$. We show that every braided $N$-Ising fusion category is
prime and also that there exists a slightly degenerate $N$-Ising braided fusion
category for all $N > 2$. We also prove a structure result for braided
extensions of a rank 2 pointed fusion category in terms of braided $N$-Ising
fusion categories.
