Group varieties not closed under cellular covers and localizations (Q1676212)

Let $G$ be a group. The group $H$ is called a cellular cover of $G$ if there exists a homomorphism $e:H\rightarrow G$ such that for every homomorphism $\phi:H\rightarrow G$ there exists precisely one endomorphism $\bar{\phi}:H\rightarrow H$ such that $\bar{\phi}e=\phi$ (equivalently, if the map $e_{*}:\text{Hom}(H,H)\to \text{Hom}(H,G)$ given by $\bar{\phi}\mapsto \bar{\phi}e$ is bijective). \par It is known that every surjective cellular cover $H$ of $G$ satisfies $H/Z(H)\cong G/Z(G).$ In particular, if $G$ is nilpotent of class $c$, then also $H$ is nilpotent of class $c$. This can be rephrased by saying that the variety of nilpotent groups of class $c$ is closed under cellular covers. \par The authors construct $2^{\aleph_0}$ varieties that are not closed under cellular covers. This answers a question of \textit{R. Göbel} [Forum Math. 24, No. 2, 317--337 (2012; Zbl 1281.20065)]. Also, examples for varieties without a finite basis for their laws and which are not closed under cellular covers are given. The constructions are heavily based on Ol'shanskij's construction of groups of prime exponent $p$ with all proper subgroups of order $p$ (see [\textit{A. Yu. Ol'shanskij}, Geometry of defining relations in groups. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.20019)]), and their universal perfect covers and other extensions. \par Dually, the group $H$ is a localization of $G$ if there exists a homomorphism $e:G\rightarrow H$ such that the map $\text{Hom}(H,H)\to \text{Hom}(G,H)$ given by $\bar{\varphi}\mapsto e\bar{\varphi}$ is bijective. Using similar techniques, $2^{\aleph_0}$ varieties are constructed that are neither closed under cellular covers nor under localization.