Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie group. (See, e.g., the answer to this question.)
It is not difficult to construct examples of $(M,g)$ whose $G$ has infinitely many connected components, but they all share the property that the fundamental group is infinite. (See, e.g., this paper.)
Question: Suppose that $M$ is simply connected. Can $G$ still have infinitely many connected components? If so, then what about if $M$ is not just simply connected but also homogeneous?
Thank you in advance for your help.