  Let $C$ be a smooth projective curve of genus $2$. Following a method by O'
Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$
of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of
degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition
theorem by Beilinson, Bernstein, Deligne one can write the cohomology of
$\tilde{\mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of
$\mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We
use this splitting to compute the intersection cohomology of
$\mathcal{M}_{Dol}^G$ and prove that the mixed Hodge structure on it is
actually pure, in analogy with what happens to ordinary cohomology in the
smooth case of coprime rank and degree.
