  We discuss the Berezin transform, a Markov operator associated to
positive-operator valued measures (POVMs). We consider the class of so-called
orbit POVMs, constructed on the quotient space $\Omega = G/K$ of a compact
group $G$ by its subgroup $K$. We restrict attention to the case where $(G, K)$
is a Gelfand pair and derive an explicit formula for the spectrum of the
Berezin transform in terms of the characters of the irreducible unitary
representations of $G$. We then specialize our results to the case study $G =
\text{SU}(2)$ and $K \simeq S^1$, and find the spectra of orbit POVMs on $S^2$.
We confirm previous calculations by Zhang and Donaldson of the spectrum of the
standard quantization of $S^2$ coming from K\"ahler geometry. Then, we make a
couple of conjectures about the oscillations in the sequence of eigenvalues,
and prove them in the simplest case of second-highest weight vector. Finally,
for low weights, we prove that the corresponding orbit POVMs on $S^2$ violate
the axioms of a Berezin-Toeplitz quantization.
