Example 2

Let E be a subgroup of F stable under the action of G. Denote by $F'$ the quotient of F by E, by $v: F\rightarrow F'$ the canonical homomorphism, and by $\tau ': G\rightarrow$ Aut(F$'$) the action of G on $F'$ characterized by

for $g\in G$ and $f\in F$. Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$-extension of G by F. Then $\iota (E)$ is a normal subgroup of Γ, and the $\tau '$-extension $v_*(\mathscr{E})$ of G by $F'$ is isomorphic to the extension $(\Gamma /\iota (E), \iota , \pi )$, where $\overline{\iota}$ and $\overline{\pi}$ are the group homomorphisms deduced from $\iota$ and $\pi$ by passing to the quotients. By this isomorphism, the canonical homomorphism $\varphi$ associated with $v_*(\mathscr{E})$ corresponds to the canonical homomorphism from Γ to $\Gamma /\iota (E)$.