The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar connected Cayley graph). Results of Tucker (https://doi.org/10.1016/0095-8956(84)90031-5) imply that there are only finitely many groups $G$ of non-orientable genus $\overline{\gamma}(G)=k$ for every $k>3$. Clearly, there are infinitely many groups with $\overline{\gamma}(G)=0$. Moreover, using results of Riskin (https://www.sciencedirect.com/science/article/pii/S0012365X0000193X) one can show that $\overline{\gamma}(\mathbb{Z}_p\times \mathbb{Z}_{p^k})=3$ for every prime $p\geq 5$ and positive integer $k$. One can see that $\mathbb{Z}_3\times\mathbb{Z}_3$ has non-orientable genus $1$, and the quaternions have non-orientable genus $2$. I still wonder:
Question 1: Are there infinitely many groups of non-orientable genus $1$, i.e., they have a connected Cayley graph embeddable in the projective plane, but no planar one?
Question 2: Are there infinitely many groups of non-orientable genus $2$, i.e., they have a connected Cayley graph embeddable in the Klein bottle, but none embedabble in the projective plane?
