Let $f$ be a polynomial of degree 5 over $\mathbb F_p$ which has no multiple roots in $\overline{\mathbb F_p}$. ($p$ is prime and large enough. We can assume $p>100$).
Is it possible to find a such $f$ such that $$f^{(p^2-1)/2}(x)=s_0(x)+x^{p^2}s_1(x)+x^{2p^2}s_2(x)$$ where $s_0,s_1,s_2$ of degree $<p(p-1)$.
Note 1: For all such polynomials, there is always such $s_2$ because of the degrees of polynomials. So the problem is actually depending to $s_0$ and $s_1$.
Note 2: Computational, I couldn't find any such $f$.
The other version of this question (not same but this one gives that the other one exists)
Is it possible to find a such $f$ such that $$f^{(p-1)/2}(x)=s_0(x)+x^{p}s_1(x)+x^{2p}s_2(x)$$ where $s_0,s_1,s_2$ of degree $<p-3$.
Similar notes here too.