  Let $k$ be an algebraically closed field. Consider a reductive group $G$ over
$k$. Let $X$ be a projective variety over $k$ with a $G$-action and let $L$ be
a very ample $G$-linearized line bundle on $X$. Suppose that $L$ descends to
the GIT quotient of $X$ by $G$. If $L$ satisfies the property $N_p$ one can ask
if its descent also has $N_p$ property. In this article, we show this is the
case under certain conditions. We then apply our results to some cases of
interest. As a consequence of our results, we show that if $G$ is a finite
group and $L$ satisfies $N_p$ property and its descent satisfies $N_0$ property
then it satisfies $N_p$ property as well under suitable conditions.
