Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$ $$ \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} f(g\gamma g^{-1}) d \mu_\gamma(g) d \nu(\gamma),$$ where $G_\gamma$ is the centralizer of $\gamma$ in $G$ and $\mu_\gamma$ is a measure on $G/ G_\gamma$? Btw., what would be the image of the operator $P : C_c(G) \rightarrow ?$, where $$ Pf(\gamma) = \int_{G / G_\gamma} f(g\gamma g^{-1}) d \mu_\gamma(g).$$ Do the conjugacy classes come with a natural topology? What is the relation to irreducible representations?

Paul Garrett has adressed the case of a reductive group, but I am interested in more general groups.

A more general question: Given an action of $G$ on a measurable space $X$, when does there exist a measure $\mu$ such that its translates $\mu_g$, where $\mu_g(A) =\mu(g^{-1}A)$, are equivalent measures?

Mark Schwarzmann has adressed this question in the comments stating that for continuous action of amenable groups on compact metrizable spaces, we actually obtain invariant measure, meaning that $\mu(gA)= \mu(A)$. I am asking for less in more general context. Note: That exy=istence (not uniqueness) of quaiinvariant measure is true for transcendental actions (=one orbit) and also when each orbits are dense + "extra condition".