Relative covering groups

Let nu and omega be two varieties of groups defined by the sets of laws V and W, respectively. We introduce the concept of a omega-nu-covering group of a given group and show that every two omega-nu-covering groups of a given group in omega are nu-isologic. Also, in the case nu subset of or equal to omega, we show the existence of such groups for a finite nu-perfect group in omega, and also that every automorphism of a finite nu-perfect group G in w may be lifted to an automorphism of a omega-nu-covering group of G. Finally we show that if G is in omegaboolean ANDnu, then all omega-nu-covering groups of G are Hopfian.