 Solution

Pass to the effective subgroup $\overline{\Gamma }\subset PS{L}_2(\mathbb{Z})$. Its index is $\mu$: adjoining $-I$ halves the original index precisely when $-I\notin \Gamma$. The natural map of compactified modular curveshas degree $\mu$, since a generic orbit has that many lifts. It branches only over the order-two elliptic orbit, the order-three elliptic orbit, and the modular cusp. Away from these, the modular quotient charts are genuine unramified covering charts.

Over $i$, a lift with its own effective stabilizer of order two is unramified; there are ${\nu }_2$ such lifts. Every other lift has ramification index two. As the sum of all ramification indices in that fibre is $\mu$, their number is $(\mu -{\nu }_2)/2$ and their contribution to the ramification divisor is $(\mu -{\nu }_2)/2$. Similarly, over $\rho$ there are ${\nu }_3$ unramified elliptic lifts and $(\mu -{\nu }_3)/3$ lifts of ramification index three, contributing $2(\mu -{\nu }_3)/3$.

At a modular cusp of width $h$, the local parameters are $q_h=e^{2\pi i\tau /h}$ upstairs and $q=e^{2\pi i\tau }$ downstairs. Thus $q=q_h^h$ and the ramification index is $h$. Summing the widths gives $\mu$, so all ${\nu }_{\infty }$ modular cusps contribute $\mu -{\nu }_{\infty }$ to ramification. The Riemann-Hurwitz formula now givesThereforeThe effective index and the projective modular cusp widths ensure that the formula also applies when the original subgroup does not contain $-I$; no spurious factor of two is introduced.