  We derive a formula for the virtual class of the moduli space of rubber maps
to $[\mathbb{P}^1/G]$ pushed forward to the moduli space of stable maps to
$BG$. As an application, we show that the Gromov-Witten theory of
$[\mathbb{P}^1/G]$ relative to $0$ and $\infty$ are determined by known
calculations.
