Let $H,K \le G$ and consider the set of double cosets $ H \backslash G / K$. If we have $H' \le H$, is there a formula expressing $| H' \backslash G / K|$ in terms of $|H \backslash G / K|$, $|H:H'|$ and other similar quantities? It may be assumed that $K \le H$ if necessary.
The motivation for this question is computing the number of cusps of modular forms, as in this question. It seems to me that there should be a way to express the number of double cosets $|\Gamma_1(p) \backslash \Gamma(1) / \Gamma_{\infty}|$ in terms of $|\Gamma_0(p) \backslash \Gamma(1) / \Gamma_{\infty}|$, $|\Gamma_0(p) : \Gamma_1(p)|$ and possibly some quantity involving $\Gamma_{\infty}$. As someone with limited knowledge of group theory, it would be helpful to have a formula which allows for a reduction in this way.
For completeness, the expressions not defined in the link are $\Gamma(1) = \text{SL}_2(\mathbb{Z})$ and $\Gamma_{\infty} = \text{stab}(\infty)$.
