Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$ I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$.
I'm reading a paper with a result that implies that such groups have $H^1(G,M)=0$ for any simple $G$-module $M$. This can help me narrow down the search.

What are the finite groups $G$ such that $H^1(G,M)=0$ for any nontrivial simple $\mathbb{F}_2G$-module $M$?

 A: A finite group $G$ has $H^1(G,M)=0$ for all non-trivial simple $kG$-modules for $k$ a field of characteristic $p$ if and only if $G$ has a normal subgroup of order coprime to $p$ and index a power of $p$ [iff $G/O_{p'}(G)$ is a $p$-group, iff $G$ has a normal $p$-complement, iff $G$ is $p$-nilpotent].
For instance the non-abelian group of order 6 is like this mod 2. In other words, the order of the group does not need to be odd, but all of the evenness has to be on top. Every finite group with a cyclic Sylow 2-subgroup is like this.
The relationship between $H^1(G,M)$ and the second-layer composition factors of $P_1 = P(k)$ is Theorem A.15.11 in Doerk–Hawkes. The main claim follows from Theorem B.4.24, since, as you noticed in your followup question, groups like this have only the trivial simple module in the principal block. You will also need a better definition of block, such as Proposition IV.13.3 in Alperin's Local Representation Theory.
Generalizable statements
In case you might want to relax your requirements some day (I don't think you'll want to for this specific problem, but I've already got the references handy): In a $p$-nilpotent group, all the $p$-chief factors live up on top (of the $p'$-part) and so have to act trivially. In general, a $p$-chief factor always lies in the first block, and the vanishing of the cohomology has a bit to do with whether it is complemented. Now a module can be in the first block without being a composition factor of $P_1 = P(k)$, but in a $p$-soluble group, the $p$-chief factors must in fact be composition factors of $P_1 = P(k)$, by Theorem B.6.17.


*

*Doerk, Klaus; Hawkes, Trevor.
Finite soluble groups.
de Gruyter Expositions in Mathematics, 4. Walter de Gruyter & Co., Berlin, 1992. xiv+891 pp. ISBN: 3-11-012892-6 
MR1169099

*Alperin, J. L.
Local representation theory.
Modular representations as an introduction to the local representation theory of finite groups.
Cambridge Studies in Advanced Mathematics, 11. Cambridge University Press, Cambridge, 1986. x+178 pp. ISBN: 0-521-30660-4; 0-521-44926-X
MR860771
DOI:10.1017/CBO9780511623592
