Can there be a tower like $1 = A_0 \triangleleft \ldots \prod^t_i S_{N_i} \wr D_{m_i} \ldots \triangleleft = S_{\sum^t_i N_i m_i}$?

Question

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of $S_{\sum^t_i N_i m_i}$.

Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of order $2 m_i$.

My question: Can there be a tower like $1 = A_0 \triangleleft \ldots \prod^t_i S_{N_i} \wr D_{m_i} \ldots \triangleleft = S_{\sum^t_i N_i m_i}$?