On Heyde's theorem on the group $\mathbb{R}\times\mathbb{T}$

According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups that contain an element of order 2. While coefficients of linear forms are topological automorphisms of a group.