I am trying to find the number of ways to color a pentagon with 4 colors up to symmetries. I know that I should be using Burnside's Theorem, and so far I know that the group $D_5$ should act on the set $X=$ {$1, 2, 3, 4, 5$}, the vertices of the pentagon.
I know that $D_5$ is generated by $(12345)$ and $(14)(23)$. From here I am unsure how to proceed. The only other thing I can think of is that there should be a function $f:X \mapsto Y =$ {$Q_1$,$Q_2$,$Q_3$,$Q_4$} that sends vertices of the pentagon to the four colors.
I suppose there must be a group $X'$ of all possible colorings, that is all possible functions $f$?
Edit: Up to four colors.