Deformational symmetries of smooth functions on non-orientable surfaces

Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$ defined by the rule: $(f,h)\mapsto f\circ h$ for $f\in \mathcal{C}^{\infty}(M,\mathbb{R})$ and $h\in\mathcal{D}(M)$. Denote by $\mathcal{F}(M)$ the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ consisting of function $f:M\to\mathbb{R}$ taking constant values on connected components of $\partial{M}$, having no critical points on $\partial{M}$, and such that at each of its critical points $z$ the function $f$ is $\mathcal{C}^{\infty}$ equivalent to some homogenenous polynomial without multiple factors. In particular, $\mathcal{F}(M)$ contains all Morse maps. Let also $\mathcal{O}(f) = \{ f\circ h \mid h\in\mathcal{D}(M) \}$ be the orbit of $f$. Previously it was computed the algebraic structure of $\pi_1\mathcal{O}(f)$ for all $f\in\mathcal{F}(M)$, where $M$ is any orientable compact surface distinct from $2$-sphere. In the present paper we compute the group $\pi_0\mathcal{S}(f,\partial\mathbb{M})$, where $\mathbb{M}$ is a M\"obius band, and $\mathcal{S}(f,\partial\mathbb{M}) = \{ h\in\mathcal{D}(\mathbb{M}) \mid f\circ h = f, \ h|_{\partial \mathbb{M}} = \mathrm{id}_{\mathbb{M}}\}$ is the subgroup of the corresponding stabilizer of $f$ consisting of diffeomorphisms fixed on the boundary $\partial \mathbb{M}$. As a consequence we obtain an explicit algebraic description of $\pi_1\mathcal{O}(f)$ for all non-orientable surfaces distinct from Klein bottle and projective plane.