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Jesper Grodal and I once looked at this, cf. this answer. In particular Eilenberg and MacLane constructs a universal obstruction in $H^3(\text{Out}(G);Z(G))$ for the extension $1\rightarrow G\rightarr …

The name metacyclic is normally used for a group which is cyclic-by-cyclic (ie. a group $G$ with a cyclic normal subgroup $N$ such that $G/N$ is also cyclic). I will therefore refer to a finite group …

(Too long for a comment.) A Magma computation shows that for $k=\mathbf{F}_p$ with $p$ prime the group $H^1(\operatorname{PSL}_2(k);k^3)$ equals $0$ for $p=3$ and $7\le p\le 17$ while the cohomology g …

I think the following is an example of $Ш(G,M(G,H,\Bbb{F}_2))\neq 0$: Take $G=A_4$ and $H$ of order $2$. Then $M$ has dimension $5$ and a (computer) calculation shows that $Ш(G,M(G,H,\Bbb{F}_2))$ has …

A theorem of Borovik, Pyber and Shalev (Corollary 1.6) shows that the number of subgroups of a group $G$ of order $n=\lvert G\rvert$ is bounded by $n^{(\frac{1}{4}+o(1)) \log_2(n)}$. This is essential …

I dont know a proof (other than by inspecting the list of degrees) in the irreducible case. However in the reducible case, a finite real reflection group is not determined up to isomorphism by its deg …