  The aim of this paper is to study Sasakian immersions of compact Sasakian
manifolds into the odd-dimensional sphere equipped with the standard Sasakian
structure. We obtain a complete classification of such manifolds in the
Einstein and $\eta$-Einstein cases when the codimension of the immersion is
$4$. Moreover, we exhibit infinite families of compact Sasakian
$\eta$--Einstein manifolds which cannot admit a Sasakian immersion into any
odd-dimensional sphere. Finally, we show that, after possibly performing a
${\mathcal {D}}$-homothetic deformation, a homogeneous Sasakian manifold can be
Sasakian immersed into some odd-dimensional sphere if and only if $S$ is
regular and either $S$ is simply--connected or its fundamental group is finite
cyclic.
