Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V_{ss}$ is a variety of semi-stable points). Let $V$ be a vector bundle on $X$ with a $G$-action commuting with the $G$-action on $X$. I believe that if $G$ acts effectively then there is a quotient vector bundle $V/G$ on $X/G$. Is it true? Could you give me references? Also, does all line bundles on $X/G$ arise this way?

Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V_{ss}$ is a variety of semi-stable points). Let $E$ be a vector bundle on $X$ with a $G$-action commuting with the $G$-action on $X$. I believe that if $G$ acts effectively then there is a quotient vector bundle $E/G$ on $X/G$. Is it true? Could you give me references? Also, does all line bundles on $X/G$ arise this way?