Total perfect codes in Cayley graphs

A total perfect code in a graph $\Gamma$ is a subset $C$ of $V(\Gamma)$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. We give necessary and sufficient conditions for a conjugation-closed subset of a group to be a total perfect code in a Cayley graph of the group. As an application we show that a Cayley graph on an elementary abelian $2$-group admits a total perfect code if and only if its degree is a power of $2$. We also obtain necessary conditions for a Cayley graph of a group with connection set closed under conjugation to admit a total perfect code.