A toric variety $X$ over $k$ is a variety which contains an algebraic torus ($T= \mathbb{G}_k^s$) as a dense open subset such that the action of the torus on itself extends to the whole of $X$. Slogan: Essentially toric varieties are just fattened tori with an action.

Let $X$ be an algebraic variety which contains a dense open subvariety $U$ and there is an algebraic group $G$ acting on $X$. Are there sufficient conditions when the natural action of $G$ on $U$ extends to the whole $X$?

The main cases in the scope of my interest are projective varieties (or weakened to 'proper'). Which role plays the base field $k$ in this extension problem.

Let me also remind that an action by an algebraic group or more general a group scheme $G$ on a algebraic variety $S$ is a morphism $f: G \times S \to S$ which respects group multiplication morphism $m: G \times G \to G$. Formally this assumption is equivalent to these two laws.

Note that's a copy of identical question I asked in MSE a week ago without getting an answer.