2 relative case added

Colliot-Thélène's paper Résolutions flasques des groupes linéaires connexes, J. für die reine und angewandte Mathematik (Crelle) 618 (2008) 77--133, http://www.math.u-psud.fr/~colliot/resolflsq_211107.pdf, contains the following theorem:

Theorem 5.6. Let $k$ be a field and $G$ be a connected linear algebraic $k$-group, assumed reductive when $\mathrm{char}(k)>0$. Let $S$ be a smooth $k$-group of multiplicative type. Let $p\colon Y\to G$ be a torsor over $G$ under $S$, whose fiber is trivial over the neutral element $e_G\in G(k)$. Then there exists a structure of algebraic $k$-group on $Y$ such that $p\colon Y\to G$ is a homomorphism of algebraic $k$-groups with central kernel $S$.

Corollary 5.7 says that that $\mathrm{Ext}_{k-\mathrm{gr}}(G,S)$ is in a canonical bijection with $\mathrm{Ker}[H^1(G,S)\to H^1(k,S)]$. Here $H^1(G,S)$ means étale cohomology.

See also http://arxiv.org/abs/0912.0408, Lemma 2.13, for a relative version of this theorem in characteristic 0 in the case $S=\mathbb{G}_m$. We consider a torsor $Y\to X$ under a connected linear $k$-group $G$ over a smooth $k$-variety $X$, and a torsor $Z\to Y$ under $\mathbb{G}_m$. We prove that $Z\to X$ has a structure of a torsor under some $k$-group $G_1$, a central extension of $G$ by $\mathbb{G}_m$. The class of such an extension $G_1$ is uniquely determined.

1

Colliot-Thélène's paper Résolutions flasques des groupes linéaires connexes, J. für die reine und angewandte Mathematik (Crelle) 618 (2008) 77--133, http://www.math.u-psud.fr/~colliot/resolflsq_211107.pdf, contains the following theorem:

Theorem 5.6. Let $k$ be a field and $G$ be a connected linear algebraic $k$-group, assumed reductive when $\mathrm{char}(k)>0$. Let $S$ be a smooth $k$-group of multiplicative type. Let $p\colon Y\to G$ be a torsor over $G$ under $S$, whose fiber is trivial over the neutral element $e_G\in G(k)$. Then there exists a structure of algebraic $k$-group on $Y$ such that $p\colon Y\to G$ is a homomorphism of algebraic $k$-groups with central kernel $S$.

Corollary 5.7 says that that $\mathrm{Ext}_{k-\mathrm{gr}}(G,S)$ is in a canonical bijection with $\mathrm{Ker}[H^1(G,S)\to H^1(k,S)]$. Here $H^1(G,S)$ means étale cohomology.