  Let $G$ be a finite group and $A$ be a normal subgroup of $G$. We denote by
$ncc(A)$ the number of $G$-conjugacy classes of $A$ and $A$ is called
$n$-decomposable, if $ncc(A)=n$. Set ${\cal K}_G = \{ncc(A)| A \lhd G \}$. Let
$X$ be a non-empty subset of positive integers. A group $G$ is called
$X$-decomposable, if ${\cal K}_G = X$.
  Ashrafi and his co-authors \cite{ash1,ash2,ash3,ash4,ash5} have characterized
the $X$-decomposable non-perfect finite groups for $X = \{1, n \}$ and $n \leq
10$. In this paper, we continue this problem and investigate the structure of
$X$-decomposable non-perfect finite groups, for $X = \{1, 2, 3 \}$. We prove
that such a group is isomorphic to $Z_6, D_8, Q_8, S_4$, SmallGroup(20, 3),
SmallGroup(24, 3), where SmallGroup$(m,n)$ denotes the $m$th group of order $n$
in the small group library of GAP \cite{gap}.
