  Let $G$ be a finite group, $X$ be a smooth complex projective variety with a
faithful $G$-action, and $Y$ be a resolution of singularities of $X/G$. Larsen
and Lunts asked whether $[X/G]-[Y]$ is divisible by $[\mathbb{A}^1]$ in the
Grothendieck ring of varieties. We show that the answer is negative if $BG$ is
not stably rational and affirmative if $G$ is abelian. The case when $X=Z^n$
for some smooth projective variety $Z$ and $G=S_n$ acts by permutation of the
factors is of particular interest. We make progress on it by showing that
$[Z^n/S_n]-[Z\langle n\rangle / S_n]$ is divisible by $[\mathbb{A}^1]$, where
$Z\langle n\rangle$ is Ulyanov's polydiagonal compactification of the $n$-th
configuration space of $Z$.
