Let $G$ be a finite group and let $X$ be a set on which $G$ acts transitively. Let $Y$ be any non empty subset of $X$. Then HW problem, asked to show, in $KG$ module $KX$ we have, $\frac{1}{|G|}\sum_\limits{g\in G}g\cdot\bigg(\frac{1}{|Y|}\sum_\limits{y\in Y}y\bigg)= \frac{1}{|X|}\sum_\limits{x\in X}x$.
Verifying this is nothing but an easy manipulation. But What does this equation mean? Why this is specifically written? How to see this intuitively\geometrically?
For every $x\in X$ and $y\in Y$ there exists $g\in G$ such that $y^g=x$. The number of those $g$ is $|G_x|$. Note that $|G:G_x|=|X|$. Hence, $$\frac{1}{|G||Y|}\sum_{g\in G}\sum_{y\in Y}y^g=\frac{|G_x|}{|G|}\sum_{x\in X}x=\frac{1}{|X|}\sum_{x\in X}x$$