Example 2

For $i\in \{1,2\}$, take a group $G_i$, an abelian group $F_i$, and a group homomorphism $\tau_i$ from $G_i$ to the automorphism group of $F_i$. We denote by $\tau_1\times \tau_2: G_1\times G_2\rightarrow$ Aut(F$_1\times F_2$) the homomorphism defined by

for $g_1\in G_1,g_2\in G_2,f_1\in F_1$, and $f_2\in F_2$. Let $\mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1)$ be a $\tau_1$-extension of $G_1$ by $F_1$ and $\mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2)$ a $\tau_2$-extension of $G_2$ by $F_2$. Then the triple $(\Gamma_1\times \Gamma_2, \iota_1\times \iota_2, \pi_1\times \pi_2)$ is a $\tau_1\times \tau_2$-extension of $G_1\times G_2$ by $F_1\times F_2$; it is called the exterior product of the extensions $\mathscr{E}_1$ and $\mathscr{E}_2$ and denoted by $\mathscr{E}_1\times \mathscr{E}_2$.