Recently I have questioned in here to understand the terms used for expressing Burnside's lemma $|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$. Someone explained the terms like the following way but I can not understand it.
Here a group $G$ acts on a set $X$. $X/G$ is the quotient of $X$ by $G$; that is, the set of orbits $\{G\cdot x\mid x\in X\}$, or equivalently the set of equivalence classes of $X$ where the equivalence relation is that $x\sim y$ if there is an element $g\in G$ so that $g\cdot x=y$. $X^g$ is the set of fixed points of a group element $g\in G$; i.e. $X^g=\{x\in X\mid g\cdot x=x\}$. So what the equation is saying that the umber of orbits is equal to the average number of fixed points.
I want to solve Necklace problem using Burnside's lemma .What do I need to do or know for intuitive understanding of Burnside's lemma and having capability to apply this lemma to solve the above problem?