Group varieties not closed under cellular covers and localizations

A group homomorphism e : H -> G is a cellular cover of G if for every homomorphism phi : H -> G there is a unique homomorphism (phi) over bar : H -> H such that (phi) over bare = phi. Group localizations are defined dually. The main purpose of this paper is to establish 2(N0) varieties of groups which are not closed under taking cellular covers. This will use the existence of a special Burnside group of exponent p for a sufficiently large prime p as a key witness. This answers a question raised by Gobel in [12]. Moreover, by using a similar witness argument, we can prove the existence of 2(N0) varieties not closed under localizations. Finally, the existence of 2(N0) varieties of groups neither closed under cellular covers nor under localizations is presented as well.