This may be a very dumb question but I was wondering if the following train of logic is correct:
We know a connected Lie group $G$ is isomorphic to the quotient $G\cong \tilde{G}/\Gamma$ where $\Gamma\cong \pi_1(G)$ is a discrete central subgroup of $\tilde{G}$.
We also know that an automorphism of $\tilde{G}$ that takes $\Gamma$ to $\Gamma$ will descend to the quotient and hence induce and autmorphism of $G$. Moreover, an automorphism of $\tilde{G}$ restricts to an automorphism of $\Gamma$.
It seems reasonable to assume that every automorphism of $\Gamma$ arises from an automorphism described in 2. However, is it true that every automorphism of $\Gamma$ extends to an autmorphism of $\tilde{G}$ (I suppose this is a type of rigidity result). If so then it seems that the automorphism group of $G$ is determined by the automorphism of $\Gamma$.