  Let $X$ be a smooth projective variety over a perfect field $k$ of
characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known
that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius
pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural
question to ask is whether this still holds in higher dimension. Indranil
Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that
there is always a counterexample to this over any algebraically closed field of
positive characteristic which is uncountable. However, we will produce a smooth
projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which,
restricted to each prime $p$ in a nonempty open subset of $\spec\mathbb Z$,
constitutes a counterexample over $p$. Indeed, given any split semisimple
simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$,
we will show that there exists a smooth projective homogeneous space $X_Z$ over
$\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime
$p$ in a nonempty open subset of $\spec\mathbb Z$, the restriction
$V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a
counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and
Frobenius Splitting of $G/B$ in characteristic $p$.
