  We prove that the group of isometries preserving a metric foliation on a
closed Alexandrov space $X$, or a singular Riemannian foliation on a manifold
$M$ is a closed subgroup of the isometry group of $X$ in the case of a metric
foliation, or of the isometry group of $M$ for the case of a singular
Riemannian foliation. We obtain a sharp upper bound for the dimension of these
subgroups and show that, when equality holds, the foliations that realize this
upper bound are induced by fiber bundles whose fibers are round spheres or
projective spaces. Moreover, singular Riemannian foliations that realize the
upper bound are induced by smooth fiber bundles whose fibers are round spheres
or projective spaces.
