  We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic
curves of genus $g$. In order to get cohomological information we wish to make
$\s_n$-equivariant counts of the numbers of points defined over finite fields
of this moduli space. We find recurrence relations in the genus that these
numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$
for low genus, then we can do this for every genus. Information about curves of
genus 0 and 1 is then found to be sufficient to compute the answers for
$\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results are applied to the
moduli spaces of stable curves of genus 2 with up to 7 points, and this gives
us the $\s_n$-equivariant Galois (resp. Hodge) structure of their $\ell$-adic
(resp. Betti) cohomology.
