Suppose $G$ is a finite group. For a subgroup $K\leq G$, I define $e_K\in\mathbb{C}G$ to be the standard trivial idempotent $\frac{1}{|K|}\sum_{k\in K}k$.
Fixing a subgroup $H\leq G$, and putting $$ x=\sum_{g\in [G/N_G(H)]} e_{gHg^{-1}}, $$ where my notation denotes that the sum is over a complete set of coset representatives of $G/N_G(H)$, we get a central element in $\mathbb{C}G$.
My question, is this element idempotent, or close to it? (Perhaps something like $x^2=cx$ for some constant $c$?) For instance, if $H$ is normal, then $x=e_H$, which is idempotent. But generally, I don't see a reasonable way to square this sum if there is one.
Well, just try out a small example; I'm afraid you will find that your $x$ will be quite far from being idempotent.
In the smallest case, let $G=S_3$ and $H=\langle(1\;2)\rangle$. Then you have $$\begin{align}x&=\frac{3}{2}+\frac{1}{2}\big((1\;2)+(1\;3)+(2\;3)\big)\text{, but}\\ x^2&=3+\frac{3}{2}\big((1\;2)+(1\;3)+(2\;3)\big)+\frac{3}{4}\big((1\;2\;3)+(1\;3\;2)\big)\end{align}$$