  Let $\Gamma$ be a graph with vertex set $V$, and let $a$ and $b$ be
nonnegative integers. A subset $C$ of $V$ is called an $(a,b)$-regular set in
$\Gamma$ if every vertex in $C$ has exactly $a$ neighbors in $C$ and every
vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In particular, $(0,
1)$-regular sets and $(1, 1)$-regular sets in $\Ga$ are called perfect codes
and total perfect codes in $\Ga$, respectively. A subset $C$ of a group $G$ is
said to be an $(a,b)$-regular set of $G$ if there exists a Cayley graph of $G$
which admits $C$ as an $(a,b)$-regular set. In this paper we prove that, for
any generalized dihedral group $G$ or any group $G$ of order $4p$ or $pq$ for
some primes $p$ and $q$, if a nontrivial subgroup $H$ of $G$ is a $(0,
1)$-regular set of $G$, then it must also be an $(a,b)$-regular set of $G$ for
any $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $a$
is even when $|H|$ is odd. A similar result involving $(1, 1)$-regular sets of
such groups is also obtained in the paper.
