Lifting an automorphism of a curve to characteristic zero

Question. Given a connected curve C 0 , proper and smooth over a field K of characteristic p, given a subgroup H ⊂ Aut(C 0); can we lift the pair (C 0 , H) to characteristic zero? ((We shall see that the answer is " NO " in general; for cyclic groups we conjecture that the answer is " YES " .)) Example (1) (Roquette). Consider the normalization of the completion of the curve given by the affine equation Y 2 = X p −X. For p ≥ 5 its genus equals (p−1)/2 and over an algebraically closed field #(Aut(C 0)) = 2p·(p 2 −1). For p ≥ 5 we see that g ≥ 2 and #(Aut(C 0)) > 84(g−1). By the Hurwitz bound we conclude that (C 0 , H) cannot be lifted to characteristic zero for H = Aut(C 0). Remark. By this example we knew that the Hurwitz bound #(Aut(C)) ≤ 84(g − 1) for a curve of genus g ≥ 2 does not hold in characteristic p. Stichtenoth proved: #(Aut(C)) ≤ 16·g 4. Another bound was given by B. Singh. Example (2). Take the curve defined in the previous example with p = 5; in this case g = 2. Consider automorphisms of this curve defined by β(X) = X + 1, β(Y) = Y, and γ(X) = −X, γ(Y) = 2Y.