Say $G$ is a reductive group over a field $k$. I usually take $k = \mathbb{C}$ so assume what you want about the field except maybe that its finite. If $X$ is a scheme over $k$ then a principal $G$ bundles over $X$ is a scheme $P$ together with a right action of $G$ and an equivariant projection to $X$ (with trivial action on $X$) such that $P$ is locally trivial in the etale topology. For some groups like $GL\_n,SL\_n$ and solvable groups principal bundles are locally trivial even in the Zariski topology. These are called special groups and Grothendieck classified them. 


I'm curious if $G',G''$ are special groups and $G$ fits into an exact sequence $1 \to G' \to G \to G'' \to 1$, then is it the case that $G$ is special?


There is a paper by Serre that claims this at least for $G',G''$ commutative and its supposed to be a consequence of the exact sequence $\check H^1(X,G') \to \check H^1(X,G) \to \check H^1(X,G'')$. This is \check Cech cohomology in the etale topology. You have $\check H^1(X,G') \cong \check H^1(X\_{zar}, G')$ and $\check H^1(X,G'') \cong \check H^1(X\_{zar}, G'')$ and a map $\check H^1(X\_{zar},G') \to \check H^1(X, G)$ but it seems you are still short of being able to use e.g. the 5-lemma. This can probably be deduced from Grothendieck's thm but I'm wondering if there is a direct argument. 



Principal bundles in the etale and Zariski topology and extensions of the structure group
The answer is more or less as Jason says, but the proof is very easy, and does not require any cohomological machinery. If $P \to X$ is a $G$-torsor, then $P/G' \to X$ is a $G''$-torsor, hence it is Zariski-locally trivial. By passing to a cover, we may assume that it is trivial; hence $P$ has a reduction of structure group to $G'$, that is, it comes from a $G'$-torsor $P' \to X$. Such a torsor is Zariski-locally trivial, and this completes the proof.


