Let $G$ be a finitely generated group and let $\Gamma=Cay(G, S)$ be the Cayley graph of $G$ with respect to some generating set $S$.

If there exists $S$ such that $\Gamma$ is bipartite, then $G$ has an index $2$ subgroup (e.g. generated by all words of even length). Does the converse hold? (I'd guess not, but I don't know generic invariants allowing to show that a group does not admit a bipartite Cayley graph)

notbipartite, regardless of whether $G$ has index-$2$ subgroups. $\endgroup$