Let $K=\mathbb{Q}(x,y)$ be a function field of genus at least 2, with defining equation $f(x,y)=0$ (say, absolutely irreducible and with coefficients not divisible by $p$), and let $k$ be the mod-$p$ reduced function field (arising by reducing the coefficients of $f$). I would like to conclude that $K$ has trivial automorphism group from the fact that $k$ does. Under what extra conditions is this true?

(I'm thinking of something like $g(k)=g(K)$, maybe combined with $p$ being sufficiently large. Of course, $Aut(K)$ should more generally embed into $Aut(k)$ for almost all $p$, but I'd like to have at least an explicit lower bound, e.g. depending on the genus).