  We study the transverse K\"ahler holonomy groups on Sasaki manifolds
$(M,{\oldmathcal S})$ and their stability properties under transverse
holomorphic deformations of the characteristic foliation by the Reeb vector
field. In particular, we prove that when the first Betti number $b_1(M)$ and
the basic Hodge number $h^{0,2}_B({\oldmathcal S})$ vanish, then ${\oldmathcal
S}$ is stable under deformations of the transverse K\"ahler flow. In addition
we show that an irreducible transverse hyperk\"ahler Sasakian structure is
${\oldmathcal S}$-unstable, whereas, an irreducible transverse Calabi-Yau
Sasakian structure is ${\oldmathcal S}$-stable when $\dim M\geq 7$. Finally, we
prove that the standard Sasaki join operation (transverse holonomy
$U(n_1)\times U(n_2)$) as well as the fiber join operation preserve
${\oldmathcal S}$-stability.
