Question Statement
Let's say I'm given a subgroup $G$ of the permutation group (i.e. the symmetric group) $S_n$. This group acts naturally on the set of n-tuples $T_n = \{(t_1, \ldots, t_n) | \{t_1, \ldots, t_n\} = \{1,\ldots,n\}\}$ (i.e. all permutations of $(1,\ldots, n)$), simply by applying the permutation. I am now allowed to restrict the set $T_n$ by introducing an arbitrary number of constraints $C = \{c_1, c_2, \ldots\}$, where each $c_i$ is of type $t_j < t_k$. Let $T_n|_C$ denote the subset of $T_n$ which satisfies all of these constraints.
I want that for any element from $T_n$, its orbit under the action of G contains exactly one element from $T_n|C$. Is it always possible to find a set of constraints $C$ such that this holds?
I'll give three simple examples:
1.) For any n, if $G = S_n$, it suffices to take $C = \{t_1 < t_2,\ \ t_2<t_3, \ \ t_3<t_4, \ \ \ldots, \ \ t_{n-1}<t_n\}$, since the orbit of any $t\in T_n$ is the entire set $T_n$ and $T_n|C$ contains only the element $(1,2,\ldots,n)$.
2.) If $n=5$ and $G = S_2 \times S_3$ (where for each $(g_1, g_2) \in G$, $g_1$ acts on the first two elements of $t\in T_n$ and $g_2$ acts on the last three), it suffices to take $C = \{t_1 < t_2, \ \ t_3<t_4, \ \ t_4<t_5\}$.
3.) If $n=4$ and $G$ is generated by the permutation $(12)(34)$, it suffices to take $C = \{t_1<t_2\}$.
What I've tried
The following is an attempt to prove this. It might not be helpful, so feel free to skip it.
I've tried to come up with an algorithm that always produces such a set of constraints $C$. The algorithm proceeds in steps, and in each step it adds one constraint to $C$ until (hopefully) we have found a set of constraints $C$ which satisfies the desired property.
At the start, $C = \emptyset$. In each step, we say that a constraint $t_i < t_j \notin C$ is bad if there exists some orbit $Gt$ such that $\forall t' \in (Gt \cap T_n|_C): t'_i > t'_j$. Adding one of these bad constraints $c$ would mean that one of the orbits has no elements from $T_n|_{C\cup \{c\}}$, so we fail immediately. Any constraint $t_i < t_j \notin C$ which is not bad is called good.
Now the question becomes whether one can always find a good constraint $c$ and an orbit $Gt$ such that $Gt\cap T_n|_{C\cup \{c\}} \subsetneq Gt\cap T_n|_C$. If so, we add $c$ to $C$ and continue with the next step. Since the orbits have finitely many elements, this algorithm would always terminate.