  Let $M$ be a compact two-dimensional manifold and, $f \in
C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its
Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in
\mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the
group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and by
$\mathcal{S}(f)=\{h\in\mathcal{D} \mid f \circ h = f\}$ the corresponding
stabilizer of this function. It is easy to show that each $h\in\mathcal{S}(f)$
induces a homeomorphism of $\Gamma_f$. Let also $\mathcal{D}_{\mathrm{id}}(M)$
be the identity path component of $\mathcal{D}(M)$, $\mathcal{S}'(f)=
\mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(M)$ be group of diffeomorphisms
of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of
homeomorphisms of the graph $\Gamma_f$ induced by diffeomorphisms belonging to
$\mathcal{S}'(f)$. This group is one of the key ingredients for calculating the
homotopy type of the orbit $\mathcal{O}_{f}$.
  Recently the authors described the structure of groups $G_f$ for Morse
functions on all orientable surfaces distinct from $2$-torus $T^2$ and
$2$-sphere $S^2$. The present paper is devoted to the case $M=S^{2}$. In this
situation $\Gamma_f$ is always a tree, and therefore all elements of the group
$G_f$ have a common fixed subtree $\mathrm{Fix}(G_f)$, which may even consist
of a unique vertex. Our main result calculates the groups $G_f$ for all Morse
functions $f:S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_f)$
consists of more than one point.
