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.