I am struggling to find a reference for what I imagine is considered a simple result (or is it false?!).
Let $G$ be a finite group, $\mathbb{C}G$ its group ring. For any subgroup $H\subseteq G$, the element $$p_H=\frac{1}{|H|}\sum_{h\in H}\delta^h$$ is an idempotent in the group ring such that $p_H(g)=\overline{p_H(g^{-1})}$ (this is an edit of the question). So that it is actually a projection in the $*$-algebra $\mathbb{C}G$.
Are all the projections in $\mathbb{C}G$ of the form $p_H$, or are there projections that don't come from a subgroup?
EDIT: The answer is already no for $G=C_3$ where we have the following four projections (thanks to rschwieb). They are $$\{p_{\{e\}},p_{C_3}, p_{\{e\}}-p_{C_3},0\}.$$
Perhaps the projections live in the linear span of $\{p_H:H\subset G\}$ but this goes beyond what I was looking for. The answer to the question is no.
If $C_3$ is the cyclic group of order $3$, then $\mathbb C[C_3]\cong\mathbb C^3$, which has $8$ idempotents.
If you impose the condition that the idempotent is invariant after $g\mapsto g^{-1}$, then there are four idempotents that do that: the trivial ones, plus $e=1/3(1+c+c^2)$ and $1-e$.
The latter one is definitely not of the form $p_H$.