The properties of covering and universality between the central extensions and also the structure of a covering group of perfect groups have been generalized by S.Kayvanfar and M.R.R.Moghaddam (Indag. Mathem., N.S, 8(4), 1997, 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 ${\\cal PME}(G,{\\cal V})$ 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.

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