Let $G$ be a complex semisimple group, $B$ a Borel subgroup of $G$ and $X=G/B$ the flag variety of $G$. If $G$ is simply connected, then every line bundle $L$ on $X$ can be made $G$-equivariant (see the proof of Theorem 1 here). Is this also true if $G$ isn't simply connected? By identifying $X$ with the flag variety of the universal covering group $\widetilde{G}$ of $G$, we can at least get a $\widetilde{G}$-action on $L$, but I haven't been able to figure out if this action factors through to yield a $G$-action.

More generally, can an arbitrary vector bundle $V$ on $X$ be made $G$-equivariant? If so, in how many ways?