On finite $X$-decomposable groups for $X={1‎, ‎2‎, ‎3‎, ‎4}$

Let $\mathcal {N}_G$ denote the set of all proper‎ ‎normal subgroups of a group $G$ and $A$ be an element of $\mathcal‎ ‎{N}_G$‎. ‎We use the notation $ncc(A)$ to denote the number of‎ ‎distinct $G$-conjugacy classes contained in $A$ and also $\mathcal‎ ‎{K}_G$ for the set $\{ncc(A)\ |\ A\in \mathcal {N}_G\}$‎. ‎Let $X$ be‎ ‎a non-empty set of positive integers‎. ‎A group $G$ is said to be‎ ‎$X$-decomposable‎, ‎if $\mathcal {K}_G=X$‎. ‎In this paper we give a‎ ‎classification of finite $X$-decomposable groups for $X=\{1‎, ‎2‎, ‎3‎, ‎4\}$‎.