Reading Huybrechts & Lehn's "The geometry of moduli spaces of sheaves" I am stuck with a particular statement that they make in the chapter on GIT without explanation. Namely, they say that for an algebraic $k$-group $G$ acting on a $k$-scheme of finite type $X$, the space of global sections of a $G$-linearized quasi-coherent $\mathcal{O}_X$-module $F$ naturally has the structure of a $G$-representation.
I give here the definition of $G$-linearization, it is essentialy the same one as given by Mumford in GIT: Let $G$ be an algebraic $k$-group, $X$ a $k$-scheme of finite type, and $\sigma : X \times G \rightarrow X$ a group action. Let $F$ be a quasi-coherent $\mathcal{O}_X$-module. Then a $G$-linearization of $F$ is an isomorphism of $\mathcal{O}_{G \times X}$-modules $\Phi : \sigma^* F \rightarrow p_1^*F$, where $p_1$ is the projection $X \times G \rightarrow X$, such that the following cocycle condition is sattisfied: $$ (id_X \times \mu)^*\Phi = p^*_{12}\Phi \circ (\sigma \times id_G)^*\Phi $$ where $p_{12} : X \times G \times G \rightarrow X \times G$ is the projection onto the first two factors.
I don't see how to use the isomorphism $\Phi$ to define a group action of $G$ on $H^0(X,F)$, nor how to use the cocycle condition to show that this action must be linear. Any help would be great, thanks.