$X$-decomposable finite groups for $X=\{1,m,m+1,m+2\}$

A normal subgroup $N$ of a finite group $G$ is called $n$-decomposable in $G$ if $N$ is the union of $n$ distinct $G$-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is $m$-decomposable, $m+1$-decomposable, or $m+2$-decomposable for some positive integer $m$. Furthermore, we give classification for the soluble case.