  We construct derived fundamental group schemes for Tate motives over
connected smooth schemes over fields. We show that there exists a pro affine
derived group scheme over the rationals such that its category of perfect
representations models the triangulated category of rational mixed Tate
motives. Under a hypothesis which is weaker than an integral version of the
Beilinson-Soule vanishing conjecture we show that there is an affine derived
group scheme over the integers such that its perfect representations model Tate
motives with integral coefficients. The hypothesis is for example fulfilled for
number fields. This generalizes previous non-derived constructions of
fundamental group schemes for Tate motives with rational coefficients.
