Corollary 1

Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be $\tau$-extensions of G by F, and let $\psi$ be a morphism of $\tau$-extensions from $\mathscr{E}_1$ to $\mathscr{E}_2$. Denote by $\varphi_1($resp. $\varphi_2)$ the canonical homomorphism for $u^*(\mathscr{E}_1) ($resp. $u^*(\mathscr{E}_2))$. Then there exists a unique morphism of $\tau '$-extensions from $u^*(\mathscr{E}_1)$ to $u^*(\mathscr{E}_2)$, denoted by $u^*(\psi )$, such that $\varphi_2\circ u^*(\psi ) =\psi \circ \varphi_1$.

It suffices to apply Proposition 1 to $\psi \circ \varphi_1$.

The class of the $\tau '$-extension $u^*(\mathscr{E})$ therefore depends only on the class of $\mathscr{E}$. We also denote by $u^*:$ Ex$_{\tau}(G,F)\rightarrow$ Ex$_{\tau'}(G',F)$ the mapping that sends the class of a $\tau$-extension $\mathscr{E}$ to the class of the $\tau '$-extension $u^*(\mathscr{E})$.