TITLE: $n$-vertex $3$-edge-colored graphs with exactly $6$ automorphisms which preserve edge color classes, but permute the edge colors distinctly?
QUESTION [5 upvotes]: In each of these $3$-edge-colored graphs, there are exactly $6$ automorphisms which preserve the set of edge color classes:

(These automorphisms don't necessarily map e.g. green edges to green edges, but if one green edge is mapped to e.g. a blue edge, then every green edge is mapped to some blue edge.)
Moreover, in each example, each of these $6$ automorphisms permutes the colors differently.
Question: For all $n \geq 3$, does there exist an $n$-vertex graph such that (a) we have exactly $6$ automorphisms which preserve the set of edge color classes, and (b) each of these $6$ automorphisms gives rise to a different permutation of the edge colors?

These come from partial Latin rectangles with a certain symmetry.  E.g. the one on the right comes from
$$
\begin{bmatrix}
1 & 2 & 3 \\
2 & 3 & \cdot \\
3 & \cdot & \cdot \\
\end{bmatrix}
$$
by making the entries the vertex set and adding a green edge when two entries share a row, an orange edge when they share a column, and a blue edge when they share a symbol.  This partial Latin rectangle has a trivial autotopism group and an automorphism group isomorphic to $S_3$ (which gives rise to the symmetry property of the graph).
It's possible that I might be able to reverse this construction, and construct interesting partial Latin rectangles this way.

REPLY [2 votes]: Proffering the following construction even though it feels "too simple".
Assume first that $n\equiv1\pmod3$. Consider a star shaped graph with one central vertex, and three "monocolor rays" of $(n-1)/3$ vertices emanating from it. Any graph automorphism must map the central vertex to itself, because it is the only vertex of degree three. It easily follows that a graph automorphism must permute the three rays. By construction all those 6 permutations preserve the edge-color classes.
Adding one isolated vertex covers the case $n\equiv2\pmod3$. 
If $3\mid n$ replace the central vertex with a cycle of three so that one ray begins from each of the three vertices of the cycle. If $P_1,P_2,P_3$ are the three vertices of that cycle, connect $P_1$ and $P_2$ with an edge sharing the color of the ray starting from $P_3$, and permute that cyclically.