  In this paper, which is the sequel to arXiv:1410.3742, we study the Frobenius
pushforward of the structure sheaf on the adjoint varieties in type ${\bf A}_3$
and ${\bf A}_4$. We show that this pushforward sheaf decomposes into a direct
sum of indecomposable bundles and explicitly determine this set that does not
depend of the characteristic. In accordance with the results of
arXiv:0707.0913, this set forms a strong full exceptional collection in the
derived category of coherent sheaves. These computations lead to a natural
conjectural answer in the general case that we state at the end.
