Faithful extension on finite order classes

In the particular case of finite orders, we investigate the notion of faithful extension among relations introduced in 1971 by R. Fraïssé: an order Q admits a faithful extension relative to an order P if P does not embed into Q and there exists a strict extension of Q into which P still does not embed. For most of the known order classes, we prove that if P and Q belong to a class then Q admits a faithful extension in this class. For the class of distributive lattices, we give an infinite family of orders P and Q such that P does not embed into Q and embeds in every strict extension of Q