  Let $\Gamma$ be a group which is virtually free of rank at least 2 and let
$\mathcal{F}_{td}(\Gamma)$ be the family of totally disconnected, locally
compact groups containing $\Gamma$ as a co-compact lattice.
  We prove that the values of the scale function with respect to groups in
$\mathcal{F}_{td}(\Gamma)$ evaluated on the subset $\Gamma$ have only finitely
many prime divisors. This can be thought of as a uniform property of the family
$\mathcal{F}_{td}(\Gamma)$.
