  Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$
be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be
its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under
the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the
space of smooth functions on $M,$ and $\mathcal{S}_V(f) =
\{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the
set $\pi_0\partial \mathcal{E}_V$ of connected components of the boundary of
$\mathcal{E}_V.$ Therefore we have a homomorphism $\phi:\mathcal{S}(f)\to
\mathrm{Aut}(\pi_0\partial \mathcal{E}_V)$. Let also $G = \phi(\mathcal{S}(f))$
be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}(\pi_0\partial
\mathcal{E}_V).$
  Suppose that the inclusion $\partial \mathcal{E}_V\subset M\setminus V$
induces a bijection $\pi_0 \partial \mathcal{E}_V\to\pi_0(M\setminus V).$ Let
$H$ be a subgroup of $G.$ We present a sufficient condition for existence of a
section $s:H\to \mathcal{S}_V(f)$ of the homomorphism $\phi,$ so, the action of
$H$ on $\partial \mathcal{E}_V$ lifts to the $H$-action on $M$ by
$f$-preserving diffeomorphisms of $M$.
  This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$
having the following property: for each critical point $z$ of $f$ the germ of
$f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to
\mathbb{R}$ without multiple linear factors.
