Proposition 3

Let G and $G'$ be groups, and let F and $F'$ be abelian groups. Let $\tau : G\rightarrow$ Aut(F)$,\tau ': G\rightarrow$ Aut(F$'$)$,u: G'\rightarrow G$, and $v: F\rightarrow F'$ be group homomorphisms such that

for $g\in G$ and $f\in F$. We write $\tau ''=\tau '\circ u$. Let $\mathscr{E}$ be a $\tau$-extension of G by F. We denote by $\varphi_u($resp. $\varphi_v,\varphi '_u,\varphi '_v)$ the canonical homomorphism corresponding to the $\tau \circ u$-extension $u^*(\mathscr{E}) ($resp. to the $\tau '$-extension $v_*(\mathscr{E})$, to the $\tau ''$-extensions $u^*(v_*(\mathscr{E}))$ and $v_*(u^*(\mathscr{E})))$. Then there exists a unique morphism $\psi$ of $\tau ''$-extensions from $v_*(u^*(\mathscr{E}))$ to $u^*(v_*(\mathscr{E}))$ such that $\varphi_v\circ \varphi_u=$ $\varphi '_u\circ \psi \circ \varphi '_v$.

We denote the $\tau \circ u$-extension $u^*(\mathscr{E})$ (resp. the $\tau ''$-extension $u^*(v_*(\mathscr{E}))$) by $(\Gamma_u, \iota_u, \pi_u)$ (resp. $(\Gamma '_u, \iota '_u, \pi '_u)$). Applying Lemma 2 of VIII, p. 287 to $\varphi_v\circ \varphi_u$, we find that there exists a group homomorphism $\psi_1: \Gamma_u\rightarrow \Gamma '_u$ such that the diagram

F $^{\iota_u}$ // $\Gamma_u^{\pi_u}$ // $G'$

$v\psi_1$

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

commutes and $\varphi '_u\circ \psi_1=\varphi_v\circ \varphi_u$. The existence of $\psi$ follows from Proposition 2 applied to $\psi_1$. Conversely, if $\psi '$ also has the desired properties, then we have $\psi '\circ \varphi '_v=\psi_1$ by Lemma 2, so $\psi '=\psi$ (Proposition 2).