Given a regular tesselation, i.e. either a platonic solid (a tesselation of the sphere), the tesselation of the euclidean plane by squares or by regular hexagons, or a regular tesselation of the hyperbolic plane.

One can consider its isometry group $G$. It acts on the set of all faces $F$. I want to define a symmetric coloring of the tesselation as a surjective map from $c:F\rightarrow C$ to a finite set of colors $C$, such that for each group element $G$ there is a permutation $p_g$ of the colors, such that $c(gx)=p_g\circ c(x)$. ($p:G\rightarrow $Sym$(C)$ is a group homomorphism).

Examples for such colorings are the trivial coloring $c:F\rightarrow \{1\}$ or the coloring of the plane as an infinite chessboard. The only nontrivial symmetric colorings of the tetrahedron, is the one, that assigns a different color to each face. For the other platonic solids there are also those colorings that assign the same colors only to opposite faces.

So my question is: Does every regular tesselation of the hyperbolic plane admit a nontrivial symmetric coloring?

I wanted to write a computer program, that visualizes those tesselations, but i didnt find a good strategy which colors should be used. So i came up with this question.