Orders for which there exist exactly six or seven groups

Much progress has been made on the problem of calculating $g(n)$ for various classes of integers $n$, where $g$ is the group-counting function. We approach the inverse problem of solving the equations $g(n)=6$ and $g(n)=7$ in $n$. The determination of $n$ for which $g(n)=k$ has been carried out by G. A. Miller for $1\le k\le 5$.