  In a graph $\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an
$(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and
every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and
$b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known
as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In
this paper we prove that, for any finite group $G$, if a non-trivial normal
subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is
also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of
integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant
b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result
involving total perfect codes is also proved in the paper.
