  The well-known Heyde theorem characterizes the Gaussian distributions on the
real line by the symmetry of the conditional distribution of one linear form of
independent random variables given another. We generalize this theorem to
groups of the form $\mathbb{R}\times F$, where $F$ is a finite Abelian group
such that its 2-component is isomorphic to the additive group of the integers
modulo $2$. In so doing, coefficients of the linear forms are arbitrary
topological automorphisms of the group. Previously, a similar result was proved
in the case when the group $F$ contains no elements of order 2. The presence of
an element of order 2 in $F$ leads to the fact that a new class of probability
distributions is characterized
