A subset $R$ of the vertex set of a graph $Γ$ is said to be $(κ,τ)$-regular if $R$ induces a $κ$-regular subgraph and every vertex outside $R$ is adjacent to exactly $τ$ vertices in $R$. In particular, if $R$ is a $(κ,τ)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(κ,τ)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $κ$ and $τ$ a pair of integers satisfying $0\leqκ\leq|H|-1$, $1\leqτ\leq|H|$ and $\gcd(2,|H|-1)\midκ$. It is proved that (i) if $τ$ is even, then $H$ is a $(κ,τ)$-regular set of $G$; (ii) if $τ$ is odd, then $H$ is a $(κ,τ)$-regular set of $G$ if and only if it is a $(0,1)$-regular set of $G$.