If $G$ is a finite group such that the Klein four group is a quotient of $G$ , then can we write $G$ as a union of three proper subgroups ?

Question

If $G$ is a finite group such that their is a normal subgroup $H$ of $G$ such that $G/H \cong V_4$ , where $V_4$ is the Klein four group , then is it true that $G$ can be written as a union of three proper subgroups ?

Answer 1

Yes: let $K,L,M$ the three proper subgroups of $V_4$, then $G=\pi^{-1}(K)\cup\pi^{-1}(L)\cup\pi^{-1}(M)$ (where $\pi$ is the quotient projection $x\mapsto x+H$) because $V_4=K\cup L\cup M$ and the counterimage of a proper subgroup under a surjective homomorphism is proper.

(In fact $\pi(\pi^{-1}(K))=K\cap\pi(G)=K$)