  Let $G$ be a simple, simply-connected algebraic group defined over
$\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$
be the subgroup of $\mathbb{F}_q$-rational points. Let $L(\lambda)$ be the
simple rational $G$-module of highest weight $\lambda$. In this paper we
establish sufficient criteria for the restriction map in second cohomology
$H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ to be an
isomorphism. In particular, the restriction map is an isomorphism under very
mild conditions on $p$ and $q$ provided $\lambda$ is less than or equal to a
fundamental dominant weight. Even when the restriction map is not an
isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(\lambda))$ in
terms of rational cohomology for $G$. We apply our techniques to compute
$H^2(G(\mathbb{F}_q),L(\lambda))$ in a wide range of cases, and obtain new
examples of nonzero second cohomology for finite groups of Lie type.
