Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In other words: the map $s$ is chosen in such a way that $\pi(s(q))=q$ for every $q\in Q$. Consider now the element $$X=\frac{1}{|Q|}\sum_{q\in Q} s(q)\in \mathbb{C}G.$$ Since $s$ is not a group homomorphism it is not necessarily true that $X$ is an idempotent.
Question: what can we still say about $X$? In particular, is it true that $Ker(X^2)=Ker(X)$? Must $X$ be semisimple?
I am in particular interested in the case where $Q=S_n$, $G$ is generated by elements $g_i, i=1,\ldots, n-1$ such that $\pi(g_i) = \sigma_i=(i,i+1)$, and the lifting $s$ is given by expressing each permutation as a reduced expression in $\sigma_i$ and sending it to the same expression with $g_i$ instead of $\sigma_i$.
