I already posed this question, but my formulation was quite erroneous and unclear so I decided to repost it (which is hopefully not against the rules).
On page 116 in Harris' book "Algebraic Geometry - A First Course" an action of a group $G$ on a projective variety $X \subset \mathbb{P}(V) \cong \mathbb{P}^n$ is defined to be linear if it lifts to the homogeneous coordinate ring $S(X)$ of $X$.
Now my problem is seeing exactly what this is supposed to mean. The term "lift" suggests, at least to me, that the action of G on $S(X)$ somehow restricts to a subset X' \subset S(X) where X' is identified in some way with the original variety.
I tried to construct such an embedding of $X$ using the identification of the underlying (n+1)-dim. K-vector space V with the homogeneous polynomials of degree 1 via $V \cong Sym^1(V) \hookrightarrow \bigoplus_{n=0}^\infty Sym^n(V) \cong K[X_0,...X_n]$
This is, however, bound to fail, and so far I have no idea how to interpret this lift of a group action. Anyways, thanks in advance.