An $n$-quasigroup $(Q,f)$ is called a G-n-quasi group iff $f=f^\sigma$ for all $\sigma\in G$, where G is a subgroup of the symmetric group of degree $n+1$ and $f^\sigma$ is defined by \[ f^igma(x_{igma_1},\dots,x_{igma_n})=x_{igma(n+1)}eftrightarrow f(x_1,\dots,x_n)=x_{n+1} \] In the paper regular permutations (Definitions 1, 2 and 3) of several classes of such $n$-quasigroups are considered and some of their properties described.