Let $\Bbbk$ be a field. Let $G$ be an affine algebraic group over $\Bbbk$. Let $X$ be a scheme over $\Bbbk$. Let $G$ act on $X$ with the action morphism $\sigma:G\times X\to X$. There are two morphisms from $G\times X$ to $X$ $$\sigma:G\times X\to X,\quad \mathrm{pr}:G\times X\to X.$$
For $\mathcal F$ a (quasi-coherent) sheaf on $X$, a linearization is an isomorhpism $$\phi:\mathrm{pr}^*\mathcal F\xrightarrow{\cong} \sigma^*\mathcal F$$ satisfying a suitable cocycle condition.
When $\mathcal F=\mathcal O_X$, there is a canonical linearization. Canonically we can identify $\mathrm{pr}^*\mathcal O_X\cong\mathcal O_{G\times X}$ and $\sigma^*\mathcal O_X\cong \mathcal O_{G\times X}$ and $\phi:\mathrm{pr}^*\mathcal O_X\to \sigma^*\mathcal O_X$ under these identification is $\mathrm{id}:\mathcal O_{G\times X}\to \mathcal O_{G\times X}$.
When $\mathcal F=\Omega_{X/\Bbbk}$, there should also be a canonical choice of $\phi:\mathrm{pr}^*\Omega_{X/\Bbbk}\to\sigma^*\Omega_{X/\Bbbk}$. However it is unclear to me at the moment.
Can you describe what is the canonical $\phi:\mathrm{pr}^*\Omega_{X/\Bbbk}\to \sigma^*\Omega_{X/\Bbbk}$?
I think that the idea here is that the descent datum for the differentials should come from pulling back differential forms along the action of group elements, using that every morphism of schemes $f \colon X \to Y$ relative to some base scheme $S$ induces a morphism of quasicoherent $\mathcal{O}_X$-modules $f^* \Omega_{Y/S} \to \Omega_{X/S}$.
To answer the question, we consider the morphism $$ \alpha \colon G \times X \to G \times X, \qquad (g, x) \mapsto (g, g.x). $$ This is an isomorphism of $G$-schemes (where on both sides we use the projection to the first factor to give the structure morphism) that can be viewed as the action of the 'universal element of $G$' on $X$. Now $\alpha$ induces an isomorphism of quasicoherent $\mathcal{O}_{G \times X}$-modules $\alpha^* \Omega_{G \times X/G} \to \Omega_{G \times X/G}$. After identifying $\Omega_{G \times X/G} \cong \mathrm{pr}_2^* \Omega_{X/k}$ and noting that $\mathrm{pr}_2 \circ \alpha = \sigma$ this gives the desired descent datum.
I think it is also instructive to compare this to the special case when $G$ is a finite group.