  If $G$ is a presheaf of groupoids on a small site, and $A$ is a sheaf of
abelian groups, we prove that the sheaf cohomology group $H^2 (BG, A)$ is in
bijection with a set of central extensions of $G$ by $A$. We use this result to
study the motivic cohomology of the Nisnevich classifying space $BG$, when $G$
is a presheaf of groups on the smooth Nisnevich site over a field, and
particularly when $G = PGL_{n}$. Finally, we show that, when $p$ is an odd
prime, the Chow ring of the classifying space of $PGL_{p}$ injects into the
motivic cohomology of the Nisnevich classifying space $BPGL_{p}$, over any
field of characteristic zero containing a primitive $p^{th}$ root of unity.
