Let $p:S^3 \to S^2$ be the Hopf fibration which is a result of the standard action of $S^1$ on $S^3$.

Is there a $2$ dimensional vector bundle $\tilde{p}:E \to S^2$ such that $S^3\subset E$ and $\tilde{p}_{|S^3}=p$?

Is there a vector bundle $E$ as above with the following extra condition:The total space $E$ can be acted by $S^1$ with linear isomorphism and this action would be the extension of the standard action of $S^1$ on $S^3$?

Note: One can ask the same question for such extension of an arbitrary principal bundle to an equivariant vector bundle of arbitrary dimension.