Example 1

Let $j:\{1\} \rightarrow F$ be the canonical injection. The semitrivial extension $\mathscr{I}_{\tau}$ is isomorphic to $j_*((G, i$, Id$_G))$, where $i:\{e\} \rightarrow G$ is the canonical injection. Let $c: F\rightarrow F$ be the constant homomorphism $f\mapsto 1$, and let $\mathscr{E}$ be a $\tau$-extension. The $\tau$-extension $c_*(\mathscr{E})$ is also isomorphic to $\mathscr{I}_{\tau}$.