  Let $X$ be an algebraic curve with an action of a finite group $G$ over a
field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits
$G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In
particular, this generalizes the result of K\"{o}ck and Tait for hyperelliptic
curves. We discuss also converse statements and tie this problem to lifting
coverings of curves to the ring of Witt vectors of length $2$.
