$\newcommand{\O}{\mathcal{O}} \newcommand{\P}{\mathbb{P}}$ Suppose one is given a vector bundle $E \to X$ and a $G$ action on $X$. Then can one show the nonexistence of a $G$-equivariant structure on the vector bundle $E$ by showing the nonexistence of a $G$-action on a certain vector space?
For example, we have the canonical action of $PGL(n+1)$ on $\P^n$ given by descending the action of $GL(n+1)$ on $\P^n$. $O(1)$ can be viewed as a subbundle of $\P^n \times \mathbb{C}^{n+1}$ of points of the form $([v],v)$. There is an obvious $GL(n+1)$ equivariant structure send$g([v],v)=([gv],gv)$. I think that one should be able to show the nonexistence of an equivariant structure by showing the nonexistence of an action of $PGL(n+1)$ on $\mathbb{C}^{n+1}$.