Heyde proved that a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear statistic given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables $ξ_1$, $ξ_2$ with values in a group $X=\mathbb{R}^n\times D$, where $D$ is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear statistic $L_2 = ξ_1 + δξ_2$ given $L_1 = ξ_1 + ξ_2$, where $δ$ is a topological automorphism of $X$ such that ${Ker}(I+δ)=\{0\}$.