I am quite new to representation theory and I reading Serre's Linear Representation of Finite Groups. In the first and second chapter, one trick he uses quite often is the substitution \begin{equation} h^0=\frac{1}{|G|}\sum_{t\in G}\rho^2_{t^{-1}}h\rho^1_{t}, \end{equation} where $\rho^j:G\to GL(V^j)$ are representations on $V^j$, $h:V_1\to V_2$ is linear, and $|G|$ is the order of the group. For instance he uses this when proving every representation is the direct sum of irreducible representations, and when proving Schur's lemma.
I wonder whether there is something behind this powerful trick. To me it seems like a method to gauge the torsion/ tension between the two representations, and then averaging over $G$, but I am not sure.
Thanks!