Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.

We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic.

In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group $L_2\left(49\right)$. As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to $L_2\left(49\right)$ except $L_2\left(49\right)·2^2$. Also, we prove that if $M$ is an almost simple group related to $L_2\left(49\right)$ except $L_2\left(49\right)·2^2$ and $G$ is a finite group such that $\left|G\right|=\left|M\right|$ and $\Gamma \left(G\right)=\Gamma \left(M\right)$, then $G\cong M$.

We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find a less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders $3$ and $5$.