Finitely generated varieties of distributive doublep-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebraswhose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and showsthat any unavoidable direct factor G is a Boolean group with at most eight elements.