  Covariant integral quantizations are based on the resolution of the identity
by continuous or discrete families of normalised positive operator valued
measures (POVM), which have appealing probabilistic content and which transform
in a covariant way. One of their advantages is to allow to circumvent problems
due to the presence of singularities in the classical models. In this paper we
implement covariant integral quantizations for systems whose phase space is
$\mathbb{Z}\times\,\mathbb{S}^1$, i.e., for systems moving on the circle. The
symmetry group of this phase space is the discrete \& compact version of the
Weyl-Heisenberg group, namely the central extension of the abelian group
$\mathbb{Z}\times\,\mathrm{SO}(2)$. In this regard, the phase space is viewed
as the right coset of the group with its center. The non-trivial unitary
irreducible representation of this group, as acting on $L^2(\mathbb{S}^1)$, is
square integrable on the phase space. We show how to derive corresponding
covariant integral quantizations from (weight) functions on the phase space
{and resulting resolution of the identity}. {As particular cases of the latter}
we recover quantizations with de Bi\`evre-del Olmo-Gonzales and
Kowalski-Rembielevski-Papaloucas coherent states on the circle. Another
straightforward outcome of our approach is the Mukunda Wigner transform. We
also look at the specific cases of coherent states built from shifted
gaussians, Von Mises, Poisson, and Fej\'er kernels. Applications to stellar
representations are in progress.
