  Let $M$ be a smooth manifold and $\mathcal{F}$ a Morse-Bott foliation on $M$
with a compact critical manifold $\Sigma$. Denote by $\mathcal{D}(\mathcal{F})$
the group of diffeomorphisms of $M$ leaving invariant each leaf of
$\mathcal{F}$. Under certain assumptions on $\mathcal{F}$ it is shown that the
computation of the homotopy type of $\mathcal{D}(\mathcal{F})$ reduces to three
rather independent groups: the group of diffeomorphisms of $\Sigma$, the group
of vector bundle automorphisms of some regular neighborhood of $\Sigma$, and
the subgroup of $\mathcal{D}(\mathcal{F})$ consisting of diffeomorphisms fixed
near $\Sigma$. Examples of computations of homotopy types of groups
$\mathcal{D}(\mathcal{F})$ for such foliations are also presented.
