  Let $M$ be a connected orientable compact surface, $f:M\to\mathbb{R}$ be a
Morse function, and $\mathcal{D}_{\mathrm{id}}(M)$ be the group of
difeomorphisms of $M$ isotopic to the identity. Denote by
$\mathcal{S}'(f)=\{f\circ h = f\mid h\in\mathcal{D}_{\mathrm{id}}(M)\}$ the
subgroup of $\mathcal{D}_{\mathrm{id}}(M)$ consisting of difeomorphisms
"preserving" $f$, i.e. the stabilizer of $f$ with respect to the right action
of $\mathcal{D}_{\mathrm{id}}(M)$ on the space
$\mathcal{C}^{\infty}(M,\mathbb{R})$ of smooth functions on $M$. Let also
$\mathbf{G}(f)$ be the group of automorphisms of the Kronrod-Reeb graph of $f$
induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is an
important ingredient in determining the homotopy type of the orbit of $f$ with
respect to the above action of $\mathcal{D}_{\mathrm{id}}(M)$ and it is trivial
if $f$ is "generic", i.e. has at most one critical point at each level set
$f^{-1}(c)$, $c\in\mathbb{R}$.
  For the case when $M$ is distinct from $2$-sphere and $2$-torus we present a
precise description of the family $\mathbf{G}(M,\mathbb{R})$ of isomorphism
classes of groups $\mathbf{G}(f)$, where $f$ runs over all Morse functions on
$M$, and of its subfamily $\mathbf{G}^{smp}(M,\mathbb{R}) \subset
\mathbf{G}(M,\mathbb{R})$ consisting of groups corresponding to simple Morse
functions, i.e. functions having at most one critical point at each connected
component of each level set.
  In fact, $\mathbf{G}(M,\mathbb{R})$, (resp.
$\mathbf{G}^{smp}(M,\mathbb{R})$), coincides with the minimal family of
isomorphism classes of groups containing the trivial group and closed with
respect to direct products and also with respect to wreath products "from the
top" with arbitrary finite cyclic groups, (resp. with group $\mathbb{Z}_2$
only).
