Classifying space of non-connected Lie groups

Question

Let $G$ be a Lie group and $G_1$ the connected component of the identity. Is there anything we can say about the classifying groups $BG$ and $B(G_1)$? I suspect no since we could take a discrete group and $B(G_0)$ will be in general very different to $BG$. But possibly something like: the free parts of the homology groups of $BG$ and $B(G_1)$ are the same holds.