Sunlet factors for Cartesian products of cycles

A sunlet is a cycle with a pendant edge attached at each vertex of the cycle. For the bipartite toroidal grid graphs $C_{2n}\mathrm{\square }{C}_{2n}$, factorizations into sunlets are given by homomorphisms from disjoint unions of $s$ copies of a sunlet for $s\in \{1,n,n^2\},n\ge 3$ such that edges are mapped bijectively.