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?

Let $B_{(H)}=\{x\in B:G_{x}\quad is\quad conjugate\quad to\quad H\}$, then every connected component of $B_{(H)}$ is a closed submanifold. Consider the isotropy representation: for any $x$, $y$ contained in the same connected component of $B_{(H)}$ we have representations $$\rho_{x}:G_{x}\longrightarrow GL(R^{k})$$ and $$\rho_{y}:G_{y}\longrightarrow GL(R^{k})$$ is it true that $\rho_{x}=\rho_{y}$?