Let $H$ be a finite group. We write ${{\mathbb{C}}}^{*n}$ for the $n$-dimensional complex torus $({{\mathbb{C}}}^*)^n$. We have a short exact sequence $$ 0\to {{\mathbb{Z}}}^n\to {{\mathbb{C}}}^n\to{{\mathbb{C}}}^{*n}\to 1,$$ which gives a connecting isomorphism $$ \Delta\colon {\rm Hom}(H,{{\mathbb{C}}}^{*n})\to H^2(H,{{\mathbb{Z}}}^n). $$

Let $\varphi\colon H\to {{\mathbb{C}}}^{*n}$ be a homomorphism. We obtain a central group extension $$ 0\to {{\mathbb{Z}}}^n\to E(\varphi)\to H\to 1$$ corresponding to $\Delta(\varphi)\in H^2(H,{{\mathbb{Z}}}^n)$. Namely, $$ E(\varphi)=H\times_{{{\mathbb{C}}}^{*n},\varphi} {{\mathbb{C}}}^n.$$

Now let $\sigma\in {\rm Aut}({{\mathbb{C}}})$ be an automorphism (not necessarily continuous). We define an action of $\sigma$ on ${{\mathbb{C}}}^{*n}$ by $\sigma(z_1,\dots,z_n)=(\sigma(z_1),\dots,\sigma(z_n))$. We obtain a new homomorphism $$ \sigma\varphi:=\sigma\circ\varphi\colon\ H\to {{\mathbb{C}}}^{*n}$$ and a new extension $$ 0\to {{\mathbb{Z}}}^n\to E(\sigma\varphi)\to H\to 1,$$ which is not isomorphic to $E(\varphi)$ when $\sigma\varphi\neq\varphi$.

**Question.** Can it happen that $E(\sigma\varphi)$ is not isomorphic to $E(\varphi)$ as *groups* (and not only as group extensions)?

*Motivation*: I am trying to construct a homogeneous space $G/H$ of a connected linear ${{\mathbb{C}}}$-group $G$,
such that the topological fundamental groups of $\tau(G/H)$ and of $G/H$ are not isomorphic.