What corresponds to the operation of taking traces in of the Fourier transformation on a finite group? I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
Let $G$ be a finite non-abelian group, $\hat{?}:L_2(G)\to \hat L_2(\hat G)$ be its Fourier transformation and $\check{?}:\hat L_2(\hat G)\to L_2(G)$ be the inverse Fourier transformation. The space $L_2(G)$ carries the standard norm induced by the inner product $\langle x,y\rangle=\sum_{g\in G}x(g)\cdot\overline{y(g)}$. The norm on $\hat L_2(\hat G)$ is normalized so that the Fourier transformations $\hat?$ is an isometry.
Let $P:\hat L_2(\hat G)\to \hat L_2(\hat G)$ be the function assigning to a sequence of matrices $(M_\alpha)_{\alpha\in\hat G}$ in $\hat L_2(\hat G)$ the sequence $(\frac 1{\dim(\alpha)}Tr(M_\alpha)\cdot\mathrm{Id}_{\alpha})_{\alpha\in\hat G}$ of normalized traces of those matrices multiplied by the identity matrices. So, $P$ is a linear projection of $\hat L_2(\hat G)$ onto its $|\hat G|$-dimensional subspace.

Question 1. What is the norm of a sequence $(M_\alpha)_{\alpha\in \hat G}\in P(\hat L_2(\hat G))$ in the Hilbert space $\hat L_2(\hat G)$? Is it equal to $(\frac1{|\hat G|}\sum_{\alpha\in\hat G}|\frac1{\dim(\alpha)}Tr(M_\alpha)|^2)^{1/2}$?

I am interested in describing the projector $A=\check{?}\circ P\circ \hat{?}:L_2(G)\to L_2(G)$. It should assign to each function $f\in L_2(G)$ some class function on $G$.

Question 2. Is $A$ the averaging of $f$ over conjugacy classes? If yes, where can I find a reference to this fact?

 A: Exercise 6.2 in Serre's book on representation theory of finite groups can be interpreted as saying that for functions $f,h$ on $G$ one has
$$\sum_{g\in G}\overline{f(g)}{h(g)} = \dfrac{1}{|G|}\sum_{\rho \in \hat G}\dim\rho\cdot \mathrm{Tr}(\hat f(\rho)^*\hat h(\rho))$$ and so the correct formula for the norm should be $$\|(M_{\rho})_{\rho\in \hat G}\|^2 = \dfrac{1}{|G|}\sum_{\rho \in \hat G}\dim\rho\cdot \mathrm{Tr}(M^*_{\rho}M_\rho))$$ where $*$ denotes the Hermitian adjoint. See Teras and CECCHERINI-SILBERSTEIN et al for this version.  Serre uses a slightly different bilnear form that agrees with this one on characters.
For the second question you are correct I will use the second orthogonality relations which says that $$\sum_{\rho\in \hat G}\overline{\mathrm{Tr}(\rho(g))}\mathrm{Tr}(\rho(h))$$ is $0$ unless $g,h$ are conjugate, in which case it is $\dfrac{|G|}{|\mathrm{Cl}(g)|}$ where $\mathrm{Cl}(g)$ is the conjugacy class of $g$.
Now $P\hat f$ is the sequence whose $\rho$-component is $$\sum_{h\in G}\frac{f(h)}{\dim \rho}\mathrm{Tr}(\rho(h))I$$.
The Fourier inversion theorem says that if $k$ is the function with $\hat k=P\hat f$, then $$k(g) =\frac{1}{|G|}\sum_{\rho\in \hat G}\dim \rho\cdot \mathrm{Tr}(\rho(g)^*\hat k(\rho))$$
$$=\frac{1}{|G|}\sum_{h\in G}f(h)\sum_{\rho\in \hat G}\overline{\mathrm{Tr}(\rho(g))}\mathrm{Tr}(\rho(h))=\sum_{h\sim g}\dfrac{f(h)}{|\mathrm{Cl}(g)|}$$ where $\sim$ is conjugacy by the second orthogonality relations, which is the average value of $f$ on the conjugacy class of $g$.
