  In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion}
associated to a finite-index and finite-depth subfactor and studied the quantum
double inclusion associated to the Kac algebra subfactor $R^H \subset R$ where
$H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite
$II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. In this article
we analyse quantum double inclusions associated to the family of Kac algebra
subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes
\cdots}_{{\text{$m$ times}}} : m \geq 1 \}$. For each $m > 2$, we construct a
model $\mathcal{N}^m \subset \mathcal{M}$ for the quantum double inclusion of
$\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes
\cdots}_{{\text{$m-2$ times}}} \}$ with $\mathcal{N}^m = ((\cdots \rtimes
H^{-2} \rtimes H^{-1}) \otimes (H^m \rtimes H^{m+1} \cdots))^{\prime \prime},
\mathcal{M} = (\cdots \rtimes H^{-1} \rtimes H^0 \rtimes H^1 \rtimes
\cdots)^{\prime \prime}$ and where for any integer $i$, $H^i$ denotes $H$ or
$H^*$ according as $i$ is odd or even. In this article, we give an explicit
description of $P^{\mathcal{N}^m \subset \mathcal{M}}$ ($m > 2$), the subfactor
planar algebra associated to $\mathcal{N}^m \subset \mathcal{M}$, which turns
out to be a planar subalgebra of $^{*(m)}\!P(H^m)$ (the adjoint of the
$m$-cabling of the planar algebra of $H^m$). We then show that for $m > 2$,
depth of $\mathcal{N}^m \subset \mathcal{M}$ is always two. Observing that
$\mathcal{N}^m \subset \mathcal{M}$ is reducible for all $m > 2$, we explicitly
describe the weak Hopf $C^*$-algebra structure on $(\mathcal{N}^m)^{\prime}
\cap \mathcal{M}_2$, thus obtaining a family of weak Hopf $C^*$-algebras
starting with a single Kac algebra $H$.
