Filling gaps of the symmetric crosscap spectrum

Every finite group $G$ acts faithfully on some non-orientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of $G$. It is known that $3$ is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps.
In this paper we obtain necessary conditions for $n$ to be a gap. According to them, the smallest value of $n$ which could be a gap is in this moment $n=699$, and there remain eight possible candidates for $n<2000$.