One of the Three Families that established the Fuyuki Holy Grail War. The family that lies at the center of events in Fate/stay night and Fate/Zero (Writer: Urobuchi Gen).
In Apocrypha, they didn't become discouraged by the fact that the Greater Grail was stolen in the Third Holy Grail War… rather, they don't seem to have that kind of mentality in the first place (if they gave up, it would only be when even their greatest creation, Illyasviel, would fail to attain the Third Magic), and they are doing everything they can with the belief that they might be able to reproduce that miracle. Accordingly, they have come to reluctantly have relationships with other magi despite their insular and unsociable nature, and the Musik family took advantage of their confused state of affairs in order to receive a portion of their homunculi craft. However, what the Einzberns taught was merely the equivalent of giving a toy to a child, so it wasn't much of a problem for them.
The Musik family concealed the fact that they were part of the Yggdmillennia clan, but naturally the Einzberns know about it. However, the Greater Grail was formed by taking apart the only successful Justeaze-model homunculus, so the Einzbern agreed to cooperate with them in case they should manage to activate one.

Fate/Apocrypha material: Fate/Apocrypha Encylopedia