Let $E\longrightarrow B$ be a vector bundle of rank $k$, then its structure group $GL(R^{k})$ acts on the fibre. Now assume that $G$ is a compact Lie group and $E\longrightarrow B$ is a $G$-equivariant vector bundle. From the definition of equivariant bundle we know that $G$ acts on each fibre of $E$ as a linear isomorphism, so that there are two actions of groups on the fibre: $GL(R^{k})$ and $G$, is it true that such two actions commute?