Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\varepsilon]/(\varepsilon^p - 1)$. Denote the quotient $X/\mu_p$ by $Y$ and the quotient morphism $X\to Y$ by $\pi$.

Let $D$ be a Weil divisor on $X$. Is the sheaf $\pi_* (\mathscr{O}_X(D))$ a direct sum of sheaves of the form $\mathscr{O}_Y(E)$ for some Weil divisors $E$ on $X$?

For an action of $\mu_n$, $n$ not divisible by $p$, we get the above statement because the push-forward decomposes according to the characters of $\mu_n$, and if we assume that there is an orbit on which $\mu_n$ acts freely, then all these summands are nonzero - hence all have rank one and are reflexive. So we need a form of ,,diagonalizability'' for $\mu_p$. I don't see how the proof for $\mu_n$ above could be translated to a proof for $\mu_p$.