An ample linearization embeds equivariantly after a positive power

By [F2] applied to $X\to \mathrm{Spec}\mathbb{C}$ there are $m\ge 1$ and global sections $s_0,\dots ,s_N$ of $L^{\otimes m}$ generating $L^{\otimes m}$ whose associated morphism is a closed immersion $X\hookrightarrow {\mathbf{P}}_{\mathbb{C}}^N$ with $\mathcal{O}(1)$ pulling back to $L^{\otimes m}$; in particular $L^{\otimes m}$ is globally generated and $\mathrm{\Gamma }(X,L^{\otimes m})\mathrm{\ne }0$.

Put $H=\mathrm{\Gamma }(X,L^{\otimes m})$, finite-dimensional by projective cohomology finiteness, and $V=H^{\ast }$. The complete-system morphism $j:X\to \mathbf{P}(V)$ exists by global generation [F3], with $j^{\ast }\mathcal{O}(1)=L^{\otimes m}$. Let $W\subseteq H$ be the span of the generating sections used for the closed immersion of step 1.1, discard linear relations, and extend a basis of $W$ to a basis of $H$. Projection to the $W$-coordinates is defined on the open $U\subseteq \mathbf{P}(V)$ where those coordinates do not all vanish, and $j(X)\subseteq U$. On each standard chart for a generating section in $W$, the map from the affine chart coordinate ring to the corresponding open of $X$ is surjective already using ratios from $W$, since the subsystem map is a closed immersion. Adding the other ratios preserves surjectivity, so $j$ is a closed immersion into $U$. Finally $X$ is projective, hence proper, and $\mathbf{P}(V)$ is separated, so $j$ is proper; its image is closed in $\mathbf{P}(V)$. Its closed immersion into $U$ therefore is a closed immersion into $\mathbf{P}(V)$ as well.

Equivariance. By [F1] the space $\mathrm{\Gamma }(X,L^{\otimes m})$ is a rational $G$-module, so its dual $V$ carries the contragredient rational structure by [F4]. The evaluation map $\mathrm{\Gamma }(X,L^{\otimes m}){\otimes }_{\mathbb{C}}{\mathcal{O}}_X\to L^{\otimes m}$ is $G$-equivariant: for a section $\sigma$, a point $x$ and $g\in G$ one has $(g\cdot \sigma )(gx)=g\sigma (x)$, because $(g\cdot \sigma )(gx)=g\sigma (g^{-1}gx)$ by the definition of the action. Hence the morphism defined by the complete linear system intertwines the actions and is $G$-equivariant for the induced action on $\mathbf{P}(V)$. The evaluation quotient also identifies $i^{\ast }\mathcal{O}(1)$ with $L^{\otimes m}$ equivariantly: the fibre of the tautological line at $i(x)$ is the evaluation line in $H^{\ast }$, and dualizing its equivariant inclusion gives exactly the equivariant evaluation quotient $H\to L_x^{\otimes m}$. With step 2.1 this gives the required $G$-equivariant closed immersion with $i^{\ast }\mathcal{O}(1)\cong L^{\otimes m}$.

The integer $m$, the finite-dimensional rational module $V$ and the $G$-equivariant closed immersion $i$ defined by the complete linear system have been produced in steps 2.1 and 3.1, and $i^{\ast }\mathcal{O}(1)\cong L^{\otimes m}$ holds by step 2.1.