Remark

Let $\mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1)$ and $\mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2)$ be $\tau$-extensions of G by F. Let $\mathscr{E}_1\mathscr{E}_2= (\Gamma , \iota , \pi )$ be the product of these extensions. Following the example of VIII, p. 289, the construction gives a surjective group homomorphism from the fiber product $\Gamma_1\times_G\Gamma_2$ to Γ whose kernel is the image of the group homomorphism $f\mapsto (\iota_1(f), \iota_2(f)^{-1})$ from F to $\Gamma_1\times_G\Gamma_2$.