TITLE: Normal Covering of a Finite Group
QUESTION [7 upvotes]: Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the main properties of finite groups with a normal covering. Anybody knows some references on this topic. There are several published papers on covering of groups by subgroups, but I like normal cover. Any comments will be highly appreciated.
Sincerely, 
F Moftakhar

REPLY [2 votes]: You can find a complete classification here:
M. Chiodo, Finitely annihilated groups, Bull. Austral. Math. Soc. 90, No. 3, 404-417 (2014).
This also contains a generalisation to arbitrary coverings of groups by proper normal finite-index subgroups; such a group is said to be Finitely Annihilated (F-A).
See in particular corollary 5.5: A finite group is F-A if and only if it has non-cyclic abelianisation.
-Maurice