= Solution
Pass to the effective subgroup $\overline\Gamma\subset PSL_2(\mathbb Z)$. Its index is $\mu$: adjoining $-I$ halves the original index precisely when $-I\notin\Gamma$. The natural map of compactified modular curves
$$
\pi:X(\Gamma)\longrightarrow X(1)\cong\mathbb P^1
$$
has 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 gives
$$
2g-2=-2\mu+\frac{\mu-\nu_2}{2}
+\frac{2(\mu-\nu_3)}{3}+\mu-\nu_\infty
=\frac{\mu}{6}-\frac{\nu_2}{2}-\frac{2\nu_3}{3}-\nu_\infty.
$$
Therefore
$$
\boxed{g=1+\frac{\mu}{12}-\frac{\nu_2}{4}-\frac{\nu_3}{3}-\frac{\nu_\infty}{2}.}
$$
The 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.
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