Approximation forte pour les vari\'et\'es avec une action d'un groupe lin\'eaire

Let $G$ be a connected linear algebraic group over a number field. Let $U \hookrightarrow X$ be a $G$-equivariant open embedding of a $G$-homogeneous space with connected stabilizers into a smooth $G$-variety. We prove that $X$ satisfies strong approximation with Brauer-Manin condition off a set $S$ of places of $k$ under either of the following hypotheses : (i) $S$ is the set of archimedean places; (ii) $S$ is a nonempty finite set and $\bar{k}^{\times}= \bar{k}[X]^{\times}$. The proof builds upon the case $X=U$, which has been the object of several works.