On finite $X$-decomposable groups for $X=\{1,2,4\}$

A normal subgroup $N$ of a finite group $G$ is called an $n$-decomposable subgroup if $N$ is a union of $n$ distinct conjugacy classes of $G$. Each finite nonabelian nonperfect group is proved to be isomorphic to $Q_{12}$, or $Z_2\times A_4$, or $G=\langle a,b,c\mid a^{11}=b^5=c^2=1,\ b^{-1}ab=a^4,\ c^{-1}ac=a^{-1},\ c^{-1}bc=b^{-1}\rangle$ if every nontrivial normal subgroup is $2$- or $4$-decomposable.