On the multiplicities of the character codegrees

Let G be a finite group and ? be an irreducible character of G, the number cod(?) = jG : Let $G$ be a finite group and $\chi$ be an irreducible character of $G$, the number $\cod(\chi) = |G: \kernel(\chi)|/\chi(1)$ is called the codegree of $\chi$. Also, $\cod(G) = \{ \cod(\chi) \ | \ \chi \in \Irr(G) \}$. For $d\in\cod(G)$, the multiplicity of $d$ in $G$, denoted by $m'_G(d)$, is the number of irreducible characters of $G$ having codegree $d$. A finite group $G$ is called a $T'_k$-group for some integer $k\geq 1$, if there exists $d_0\in\cod(G)$ such that $m'_G(d_0)=k$ and for every $d\in\cod(G)-\{d_0\}$, we have $m'_G(d)=1$. In this note we characterize finite $T'_k$-groups completely, where $k\geq 1$ is an integer.