We know $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)=\mathbb{Z}_4$. So there are two classes of $\mathbb{Z}_4$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$.
Wha are the Poincaré dual $(5-d)$-dimensional manifolds of the generators PD($a^d$) of $a^d\in H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)$, for $d=0,1,2,3,4,5$?
\begin{array}{|c|c|} \hline \text{PD}(a^d)& a^d =0 \in H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4) &a^d \neq 0 \in H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)=\mathbb{Z}_4\\ \hline d=0 &S^5?& S^5/\mathbb{Z}_4?\\ \hline d=1 &S^4? &?\\ \hline d=2 &S^3?&?\\ \hline d=3 &S^2?&?\\ \hline d=4 &S^1?&?\\ \hline d=5 & \text{a point}?& \text{a point}?\\ \hline \end{array}