Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then, $|X| \big| |G|$ by the orbit-stabilizer theorem. Let this action induce a permutation representation $\rho: G \to GL(V)$, where $V$ is a free vector space on $X$ over $\mathbb{C}$.
If $|X|=|G|$, then $\rho$ is unique up to isomorphism. Indeed, $\rho$ is the regular representation. In another world, there is only one character table for $\rho$:
$$\chi_{\rho}=\begin{cases}|X|\ \ \ g=\text{identity elements of $G$}\\ 0 \ \ \ \ \ \ \ \text{otherwise} \end{cases}.$$
Question: Is $\chi_{\rho}$ also unique when $|X| < |G|$?
After reading the first comment, I want to add for which groups the answer is yes?