  Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible
complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number
$\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$.
The set of co-degrees of all irreducible characters of $G$ is denoted by
$\rm{cod}(G)$. In this paper, we show that for a non-trivial finite group $G$,
$|\rm{Irr}(G)|=|\rm{cod}(G)|$ if and only if $G$ is isomorphic to the cyclic
group $\mathbb{Z}_2$ or the symmetric group $S_3$.
