  A very important class of homogeneous Riemannian manifolds are the so-called
normal homogeneous spaces, which have associated a canonical connection. In
this work we obtain geometrically the (connected component of the) group of
affine transformations with respect to the canonical connection for a normal
homogeneous space. The naturally reductive case is also treated. This completes
the geometric calculation of the isometry group of naturally reductive spaces.
In addition, we prove that for normal homogeneous spaces the set of fixed
points of the full isotropy is a torus. As an application of our results it
follows that the holonomy group of a homogeneous fibration is contained in the
group of (canonically) affine transformations of the fibers, in particular this
holonomy group is a Lie group (this is a result of Guijarro and Walschap).
