Abstract: A special class of Alexandrov metric spaces are the quotients X=S^n/G of the round spheres by isometric actions of compact subgroups G of O(n+1). We will consider the question of how to compute the isometry group of such X, the main result being that every element in the identity component of Isom(X) lifts to a G-equivariant isometry of the sphere. The proof relies on a pair of important results about the "smooth structure" of X.