An $n$-quasigroup $(Q,A)$ is called a G-n-quasigroup iff $A=A^\sigma$ for all $\sigma\in G$, where $G$ is a subgroup of the symmetric group of degree $n+1$ and $A^\sigma$ is defined by: \[ A^igma(x_{igma_1},\dots,X_{igma_n})=x_{igma_{n+1}}\quadext{iff}\quad A(x_1,\dots,x_n)=x_{n+1}. \] In the paper G-n-quasigroups are considered, and some of their properties described.