Lemma 3

Let $G'_1$ be a group, and let $w: G\rightarrow G'_1$ and $\tau_1: G'_1\rightarrow$ Aut(F$'$) be group homomorphisms such that $\tau '=\tau_1\circ w$. Let $\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1)$ be a $\tau_1$-extension of $G'_1$ by $F'$, and let $\varphi_1: \Gamma \rightarrow \Gamma '_1$ be a group homomorphism such that the following diagram commutes:

F $^{\iota}$ // Γ $^{\pi}$ // G

$v\varphi_1w$

${F'}^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // $G'_1$.

Then there exists a unique group homomorphism $\psi : \Gamma '\rightarrow \Gamma '_1$ such that the diagram

${F'}^{\iota'}$ // ${\Gamma'}^{\pi'}$ // G

$\psi w$

${F'}^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // $G'_1$ commutes and $\varphi_1=\psi \circ \varphi$.

For any $(f, \gamma )\in F'\times \Gamma$, denote the class of $(f, \gamma )$ in $\Gamma '$ by $(f, \gamma )$. If the group homomorphism $\psi : \Gamma '\rightarrow \Gamma '_1$ has the desired properties, then it satisfies the relations

for every $f'\in F'$ and $\gamma \in \Gamma$. Conversely, the mapping $\widetilde{\psi}$ from $F'\times_{\tau'\circ\pi}\Gamma$ to $\Gamma '_1$ given by $(f, \gamma )\mapsto \iota '_1(f)\varphi_1(\gamma )$ is a group homomorphism. Indeed, we have the relations

for $f, f'\in F'$ and $\gamma , \gamma '\in \Gamma$. The kernel of $\widetilde{\psi}$ contains the image of $j$ because $\iota '_1(v(f)) =\varphi_1(\iota (f))$ for $f\in F$, and the morphism $\psi$ deduced from $\widetilde{\psi}$ by passing to the quotient has the desired properties.