A representation of homogeneous symmetric groups by hierarchomorphisms of spherically homogeneous rooted trees are considered. We show that every automorphism of a ho- mogeneous symmetric (alternating) group is locally inner and that the group of all automorphisms contains Cartesian products of ar- bitrary finite symmetric groups. The structure of orbits on the boundary of the tree where inves- tigated for the homogeneous symmetric group and for its automor- phism group. The automorphism group acts highly transitive on the boundary, and the homogeneous symmetric group acts faith- fully on every its orbit. All orbits are dense, the actions of the group on different orbits are isomorphic as permutation groups.