  We adapt methods from quiver representation theory and Hall algebra
techniques to the counting of representations of virtually free groups over
finite fields. This gives rise to the computation of the E-polynomials of
$\mathbf{GL}_d(\mathbb{C})$-character varieties of virtually free groups. As
examples we discuss the representation theory of $\mathbb{D}_\infty$ ,
$\mathbf{PSL}_2(\mathbb{Z})$ , $\mathbf{SL}_2(\mathbb{Z})$ ,
$\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Z})$ .
