Given a locally compact group $G$, we can consider $G$ acting on its self by conjugation. Is there a quasiinvariant measure on the quotient space of $G$ with repect to multiplication from one side with an integral formula?

I will give a related example, to give an indication what I want: Let $H$, a closed subgroup act on $G$ by multiplication from the right, then there is a formula due to George Mackey $$ \int_G f(g) \rho(g) d g = \int_{G/H} \int_H f(gh) d h d \mu(gH), $$ where $\mu$ is quasiinvariant under the left multiplication by $G$ on $G/H$. The function $\rho$ can be computed from the Radon Nikodym derivative of $\rho(g) = d \mu_{g_0}/ d \mu (1)$, where $\mu_{g_0}(X) = \mu(g_0X)$.

Quasi invariance is actually exactly the property, which allows us to take the Radon Nikodym derivative. So I guess, the question is more general, what properties of an action on a topological (or measurable space) ensure the existence of quasi invariant measure. For transcendental actions, it works and this gives the second example. But conjugation actions have lots of orbits (=conjugacy classes), so what to do here? For finite group everything works of course=)