Consider a smooth tame Deligne-Mumford stack $[Y/G]$, a point $[p]$ on it with stabilizer group $H$. Is it true that every representation of $H$ can be extended to a locally free sheaf on $[Y/G]$?
Alternatively, consider a smooth scheme (or algebraic space) $Y$ with a $G$ action. Let $p$ be a point on $Y$ with a finite stabilizer group $H$. Is the restriction map from the Grothendieck group of $G$-equivariant locally free sheaves on $Y$ to the representation ring of $H$
$$K^0_G(Y)\to Rep(H)$$ $$E\mapsto E|_p$$
surjective? (An affirmative answer with possibly some additional conditions, or a negative answer with an explicit counterexample are both very welcome.)