Remark

The definition of the operation $\tau$ by formula (2) depends on the extension $\mathscr{E}'$ and the section $s'$. When $F$ is commutative, the operation $\tau$ does not depend on $s'$. For $\operatorname{Int}(s'(g)) | i'(F)$ depends then only on the coset of $s'(g)$ mod. $i'(F)$.

More generally, let $\mathscr{E} : F \to E \to G$ be an extension of $G$ by a commutative group $F$ (it is not assumed that $\mathscr{E}$ admits a section). The group $E$ operates on $F$ by inner automorphisms, this image is trivial on the image of $F$ and hence defines an operation of $G$ on $F$. If $\mathscr{E}$ admits a section, this operation is that defined by formula (2).