  Let $p:S\to S_g$ be a finite $G$-covering of a closed surface of genus $g\geq
1$ and let $B$ its branch locus. To this data, it is associated a
representation of a finite index subgroup of the mapping class group
$\operatorname{Mod}(S_g\smallsetminus B)$ in the centralizer of the group $G$
in the symplectic group $\operatorname{Sp}(H_1(S,{\mathbb Q}))$. They are
called \emph{virtual linear representations} of the mapping class group and are
related, via a conjecture of Putman and Wieland, to a question of Kirby and
Ivanov on the abelianization of finite index subgroup of the mapping class
group. The purpose of this paper is to study the restriction of such
representations to the hyperelliptic mapping class group
$\operatorname{Mod}(S_g,B)^\iota$, which is a subgroup of
$\operatorname{Mod}(S_g\smallsetminus B)$ associated to a given hyperelliptic
involution $\iota$ on $S_g$. We extend to hyperelliptic mapping class groups
some previous results on virtual linear representations of the mapping class
group. We then show that, for all $g\geq 2$, there are virtual linear
representations of the hyperelliptic mapping class group with nontrivial finite
orbits, associated to $G$-coverings of $(S_g,\iota)$ ramified over the locus of
Weierstrass points.
