I've been revisiting group cohomology, and I realized that there is something I never quite understood. Let $G$ be a finite group, and let $A$ be a $G$-module (i.e. $\mathbb{Z}[G]$-module). Then the second cohomology $H^2(G,A)$ classifies group extensions $1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1$ such that the action of $G$ on $A$ by left conjugation jibes with the given action of $G$ on $A$.
The interpretation of the first cohomology group, however, I still find somewhat elusive. On the one hand, there is the following interpretation: $H^1(G,A)$ is isomorphic to $\operatorname{Aut} ( 1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1) / \sim$ (where $\sim$ is conjugation by an element of $A$, and where $\operatorname{Aut}$ is taken in the category of group extensions of $G$ by $A$) by $\langle \rangle \mapsto (e \mapsto \langle f(e)\rangle e)$, where $f$ is the map $E\rightarrow G$, and where $\langle \rangle$ is a $1$-cocycle. This is indeed an isomorphism.
However, there is another interpretation! Some places say that there is a bijection between $H^1(G,A)$ and sections of $f:E\rightarrow G$ up to conjugation by an element of $A$! Here the problem is that I can't quite decipher if this should mean set-theoretic sections or group-theoretic sections.
Here's how I'm thinking about this: you can let $\operatorname{Aut}(E)/\sim$ act on the set of set-theoretic sections of $f$ by $\phi$ acts on $\lambda$ taking it to the section $\phi\circ \lambda$. It seems like this action is free. Furthermore, it seems like it is transitive. Indeed, if you want to take $\lambda$ to $\lambda'$ you can use the automorphism $e\mapsto \lambda'(f(e))\lambda^{-1}(f(e))e$ ($e\mapsto \lambda'(f(e))\lambda^{-1}(f(e))$ is a $1$-cocycle). So it seems like this gives a (non-canonical) bijection between $H^1(G,A)$ and the set-theoretic sections. But something is wrong -- this action takes the group-theoretic sections to group-theoretic sections! So it can't act freely and transitively on the set of set-theoretic sections...
I'm not quite sure what the resolution of this is since most sources don't distinguish between group-theoretic sections and set-theoretic sections.
Also, if you can inform me a way to think about this that relates these two interpretations with the interpretation that the first cohomology classifies torsors, I would be very happy indeed!