\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
The properties of covering and universality between the central
extensions and the structure of a covering group of perfect groups
have been generalized by S. Kayvanfar and M. R. R. Moghaddam
($1997, Indag. Math. N.S.$ 8(4), 537-542) to the variety of groups
defined by a set of outer commutator words. In this paper we
generalize the above results to any variety of groups. Then we
introduce the category ${\cal P} {\cal M} {\cal E} (G, {\cal V})$
and, using the above generalization, show that if $G$ is ${\cal
V}$-perfect, then there exists a universal object in this category
and its structure will be determined. Finally it is shown that any
two ${\cal V}$-covering groups of a ${\cal V}$-perfect group are
isomorphic and the structure of the unique generalized covering
group of an arbitrary ${\cal V}$-perfect group is introduced.
\end{document}