After introducing the class of the classifying stack, $BG$, of a (finite) group $G$ in the Grothendieck ring of algebraic stacks, I plan to present certain cohomological invariants for such a group - the Ekedahl invariants.
Then, I am going to sketch the proof that the motivic class of the classifying stack of every finite linear (or projective) reflection group is trivial. The argument is base on the study of the combinatorial properties of certain subspace arrangements $bA/b^V$ associated to a representation $V$ of $G$. This arrangement and its quotient by the canonical action of the group seems to be a key combinatorial object also in the study of the motivic class of the quotient variety $U/G$, where $U$ is the complement of the singular locus of $V/G$.
These results relate to invariant theory and to Noether’s Problem - the study of the rationality of the extension $F(G) = F(x_g : g \in G)^G$ over $F$, for a field $F$ and a finite group $G$.
(Partial joint work with Emanuele Delucchi.)