  It is shown that a countable symmetric multiplicative subgroup $G=-H\cup H$
with $H\subset\mathbb{R}_+^\ast$ is the group of self-similarities of a
Gaussian-Kronecker flow if and only if $H$ is additively
$\mathbb{Q}$-independent. In particular, a real number $s\neq\pm1$ is a scale
of self-similarity of a Gaussian-Kronecker flow if and only if $s$ is
transcendental. We also show that each countable symmetric subgroup of
$\mathbb{R}^\ast$ can be realized as the group of self-similarities of a simple
spectrum Gaussian flow having the Foias-Stratila property.
