Proof of Mackey's restriction formula for the restriction of an induced rerpesentation

Question

Let $G$ be a finite group with subgroup $H$ and suppose that $(\rho,W)$ is a representation of $G$. Then I want to prove that $$ \text{Res}_H^G(\text{Ind}_H^G(W))\cong\bigoplus_{s\in S}\text{Ind}_{H^s}^H(W^s), $$ where $S$ is the set of representatives in $H\backslash G/H$ and $H^s=sHs^{-1}\cap H$ and $(\rho^s,W^s)$ is the representation of $H^s$ defined by $\rho^s(h)=\rho(s^{-1}hs)$. Now, I know this is proven in Linear Representations of Finite Groups by Serre, but I to be honest, I don't fully understand his proof. It seems that Serre uses a different definition of the induced representation, I defined it on the vector space of functions $$\text{Ind}_H^G(W)=\{f:G\to W\colon f(hg)=\rho(h)f(g)\}$$ and then together with a $G$-action on $\text{Ind}_H^G(W)$ by $(g\cdot f)(g')=f(g'g)$. Does anyone know a reference in which this decomposition is proven?