Let $G$ be a finite group, $H, K \leq G$, and let $g_1, \ldots, g_r$ be representatives of the double cosets $K \backslash G/H$. Then
\begin{align*}
\operatorname{Res}_K^G \operatorname{Ind}_H^G \mathbf{1}_H \cong \bigoplus_{i=1}^r \operatorname{Ind}_{g_i H g_i^{-1} \cap K}^K \mathbf{1}.
\end{align*}