This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the classifying stack $\mathcal{B}G$. This has a (nonrepresentable) morphism to a point: $st:\mathcal{B}G\to P$. What do people mean when they say the "fundamental class" $[\mathcal{B}G]$ of $\mathcal{B}G$? (I don't know how homology carries over to stacks.) For example, is it true that $st_*[\mathcal{B}G]=[P]$? Or that $st^*[P]=[\mathcal{B}G]$? Are these equivalent?

My guess: It seems to me that the map $st$ should be considered as "degree" $1/|G|$. Because of this it seems like we should have $st^*[P]=|G|[\mathcal{B}G]$. I'm not sure what this would say about $st_*[P]$.

Thanks!