Functorial splitting of $l$-adic cohomology in exact sequences of group varieties

In this document we consider an exact sequence of group varieties $e\to N\to G\to Q\to e$ over an algebraically closed field. We show that for $l\mathrm{\ne }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)$ a prime there exists an isomorphism of graded ${\mathbb{Q}}_l$-algebras ${\mathrm{H}}_{\acute{\text{e}}\text{t}}^{\ast }(G,{\mathbb{Q}}_l)\cong {\mathrm{H}}_{\acute{\text{e}}\text{t}}^{\ast }(N,{\mathbb{Q}}_l){\otimes }_{{\mathbb{Q}}_l}{\mathrm{H}}_{\acute{\text{e}}\text{t}}^{\ast }(Q,{\mathbb{Q}}_l)$ that is compatible with pullback homomorphisms $f^{\ast }$ of endomorphisms $f:G\to G$ that stabilize $N$.