  In this note we continue our development of tannakizations of symmetric
monoidal infinity-categories, begun in our previous paper. The issue treated in
this paper is the calculation of tannakizations of examples of symmetric
monoidal stable infinity-categories with fiber functors. We consider the case
of symmetric monoidal infinity-categories of perfect complexes on perfect
derived stacks. The first main result especially says that our tannakization
includes the bar construction for an augmented commutative ring spectrum and
its equivariant version as a special case. We apply it to the study of the
tannakization of the stable infinity-category of mixed Tate motives over a
perfect field. We prove that its tannakization can be obtained from the
torus-equivariant bar construction of a commutative differential graded algebra
equipped with torus-action. Moreover, under Beilinson-Soule vanishing
conjecture, we prove that the underlying group scheme of the tannakization is
the conventional motivic Galois group for mixed Tate motives. The case of Artin
motives is also included.
