This article concludes the comprehensive study started in [Sz5], where the first nontrivial isospectral pairs of metrics are constructed on balls and spheres. These investigations incorporate four different cases since these balls and spheres are considered both on $2$-step nilpotent Lie groups and on their solvable extensions. In [Sz5] the considerations are completely concluded in the ball-case and in the nilpotent-case. The other cases were mostly outlined. In this paper the isospectrality theorems are completely established on spheres. Also the important details required about the solvable extensions are concluded in this paper.