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. .