Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\tilde{\times }{S}^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles ${\mathbf{K}}_0$ and ${\mathbf{K}}_1$ with common boundary $K$. Then the above foliations $\mathcal{G}$ on both ${\mathbf{K}}_0$ and ${\mathbf{K}}_1$ gives a foliation ${\mathcal{G}}^{\mathrm{\prime }}$ on $S^1\tilde{\times }{S}^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and ${\mathcal{G}}^{\mathrm{\prime }}$.