Perfect Colorings of the Infinite Square Grid: Coverings and Twin Colors

A perfect coloring (equivalent concepts are equitable partition and design) of a graph $G$ is function $f$ from the set vertices onto some finite (of colors) such that every node color $i$ has exactly $S(i,j)$ neighbors $j$, where constants, forming matrix $S$ called quotient. If an adjacency simple $T$ on colors, then covering target by cover $G$. We characterize all coverings infinite square grid, proving either orbit (that is, corresponds to under action group automorphisms) or twin colors two unifying them keeps perfect). The case separately classified.