Let $G$ be a topological group and $p:E\rightarrow X$ a $G$-equivariant topological vector bundle over the topological space $X$, where equivariant means that for any $v\in E$ and $g\in G$,
$$p(g\cdot v) = g\cdot p(v),$$
and $G$ acts between the fibres by linear isomorphisms. Then does this vector bundle always descend to a vector bundle over the topological space $X/G$ of $G$-orbits?
If not, what are the obstructions?
Cheers.