Let F ∞ be a free group on a countable set {x 1, x 2, …} and ν be a variety of groups, defined by the set of outer commutators V, in the free generators x i ' s. The paper is devoted to give the complete structure of a ν-covering of ν-perfect groups. Fur thermore necessary and sufficient conditions for the universality of a ν-central extension by a group and its ν-covering group will be presented.