  Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a
constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of
diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the
sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the
group $\pi_0\mathcal{S}(f,\partial B)$ of isotopy classes of such
diffeomorphisms. In fact, those computations hold for functions
$f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to
homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.
  Together with previous results of the second author this allows to compute
similar groups for certain classes of smooth functions $f:N\to\mathbb{R}$ on
non-orientable surfaces $N$.
