ag.algebraic geometry - Can there exist two non-equivalent equivariant actions of a group on vector bundle?

Maybe I am miss understanding the question, but it seems the answer is yes.

Take your favorite G-space, mine is $S^1$ with the $mathbb{Z}/2$-action "flip". Then consider the trivial vector bundles $S^1 times V$, where $V$ is a $G$-representation. In my favorite example $V = mathbb{R}$ can be either the trivial representation or the sign representation. Taking the diagonal $G$-action gives an equivariant action of the group on the vector bundle. They are distinct for distinct representations (at least in the $S^1$-example) yet the underlying vector bundles are the same if the representations have the same dimension (they are trivial bundles after all).