No, it does not. As a simple example, let $G\cong S_3$, $\Omega = G/C_2 \sqcup G/C_2 \sqcup G/C_3$. You can easily check that the corresponding permutation character is isomorphic to the regular character plus two copies of the trivial. In other words, $\mathbb{C}[G/1] \oplus \mathbb{C}[G/G]^{\oplus 2} \cong \mathbb{C}[G/C_3] \oplus \mathbb{C}[G/C_2]^{\oplus 2}$.
Your question can be rephrased as "does there exist a Brauer relation for $G$ in which the regular set $G/1$ enters with non-zero coefficient" (for more on Brauer relations see e.g. this MO question, as well as google). My hunch is that such a Brauer relation exists in any non-cyclic finite group, but I would have to think about it for a bit. Let me know if you need to know this.