  In previous work we described when a single geometric representation, valued
in a linear algebraic group, of the Galois group of a number field lifts
through a central torus quotient to a geometric representation. In this paper
we prove a much sharper result for systems of l-adic representations, such as
the l-adic realizations of a motive, having common "good reduction" properties.
Namely, such systems admit geometric lifts with good reduction outside a common
finite set of primes. The method yields new proofs of theorems of Tate (the
original result on lifting projective representations over number fields) and
Wintenberger (an analogue of our main result in the setting of lifting through
a central isogeny).
