ag.algebraic geometry - When do equivariant sheaves on a formal neighborhood extend?

Suppose that $X$ is a variety (in char 0) with an action of an affine algebraic group $G$. Let $Y subset X$ be a subvariety fixed by $G$--the action map agrees with projection upon restriction to $Y$. Let $widehat{Y}$ be the formal completion of $X$ along $Y$. Furthermore let $widehat{G}$ be the the completion of $G$ at the identity, viewed as a formal group. There is a restriction functor $j^*$ from the $Qcoh^G(X)$, the category of $G$-equivariant quasicoherent sheaves on $X$, to $Qcoh^{widehat{G}}(widehat{Y})$, the category of $widehat{G}$-equivariant quasicoherent sheaves on $widehat{Y}$.

1) Is this situation considered in the literature? Where?
2) What tools are available to control this functor? How might one describe the essential image?

Although curious about this general package, I specifically care about the case $G =mathbb{G}_m$.