Also in this problem, we require $F$ to be a field and $G$ to be a finite group. I know $\Delta$'s quotient is isomorphic to the trivial module. But I can't see why it's unique. In addition, I know the result hold for the case that $Char(F)=p$ and $G$ is a p-group. But I can't apply it here.
Let $H\subseteq FG$ be a submodule, such that $FG/H$ is isomorphic to the trivial module $F$. Let $\pi$ denote the composition $FG \rightarrow FG/H \rightarrow F$. Since this homomorphism is $FG$-linear, we have $$\pi(x) = \pi(x\cdot 1) = \epsilon(x)\cdot \pi(1) \quad \text{for all $x\in FG$.}$$ Surjectivity of $\pi$ implies $\pi(1)\neq 0$. Since $F$ is a field (and hence doesn't contain zero-divisors), we have $\pi(x)=0 \iff \epsilon(x)=0$. This shows $H = \ker\pi = \ker\epsilon=\Delta$. Thus, $\Delta$ is unique.