These are additions to Konstantinos' answer and was given before the edit
A Counterexample
This is inspired from here and here.
Let $G=S_3\times C_2$ and $\nu=\frac{1}{2}(\delta^{(e,1)}+\delta^{((12),1)})$.
Now define $S_0=\{(e,0),((12),0),((13),1),((23),1),((123),1),((132),1)\}$ and $S_1=G\backslash S_0$.
Consider $p_0=\mathbb{1}_{S_0}$ and $p_1=\mathbf{1}_{S_1}$.
These projections have all the properties given above.
Consider $$p_1=\delta_{((123),0)}+ \cdots.$$ Note $$\begin{align*} \Delta(p_1)&=\Delta(\delta_{((123),0)}+\cdots) \\&=\underbrace{\delta_{((132),0)}\otimes \delta_{((132),0)}}_{\in p_1\otimes p_1}+\cdots,\end{align*}$$ and so $$\Delta(p_1)\neq p_0\otimes p_1+p_1\otimes p_0.$$
A Missing Condition
I am missing the following condition. We suppose in addition that for all projections $q\in F(G)$, there exists $k\in \mathbb{N}$ such that $\nu^{\star k}(q)\neq 0$, where $$\nu\star \nu=(\nu\otimes\nu)\circ \Delta.$$ This will be added to the question.
In the classical case, where $F(G)$ is commutative, it can be shown that $p_0=\mathbf{1}_N$, where $N\lhd G$, $p_1=\mathbb{1}_{Ng}$, and $p_m=\mathbb{1}_{Ng^m}$, i.e. the $p_i$ are indicator functions on cosets of the normal subgroup $N\lhd G$. Furthermore, $G/N\cong C_d$, and so with the missing condition, in the classical case, the projections satisfy:
$$\Delta(p_i)=\sum_{k=0}^{d-1}p_{i-k}\otimes p_k.$$
I will edit the original question.