= Solution
The vector
$$
\mathbf1_X=\sum_{x\in X}x
$$
is fixed by every element of $G$, so $\mathbb C\mathbf1_X\cong\mathbb C_G$ is a <trivial representation>. The coefficient-sum subspace
$$
V=\left\{\sum_{x\in X}a_xx:\sum_{x\in X}a_x=0\right\}
$$
is also $G$-invariant, and every vector has a unique decomposition into a constant vector and an element of $V$. Hence
$$
\boxed{\mathbb C X=\mathbb C_G\oplus V.}
$$
Here $V$ is the <augmentation subrepresentation of a permutation representation>.
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