  We study the question of whether it is possible to determine a finitely
generated group $G$ up to some notion of equivalence from the spectrum
$\mathrm{sp}(G)$ of $G$. We show that the answer is "No" in a strong sense. As
the first example we present the collection of amenable 4-generated groups
$G_\omega$, $\omega\in\{0,1,2\}^\mathbb N$, constructed by the second author in
1984. We show that among them there is a continuum of pairwise
non-quasi-isometric groups with
$\mathrm{sp}(G_\omega)=[-\tfrac{1}{2},0]\cup[\tfrac{1}{2},1]$. Moreover, for
each of these groups $G_\omega$ there is a continuum of covering groups $G$
with the same spectrum. As the second example we construct a continuum of
$2$-generated torsion-free step-3 solvable groups with the spectrum $[-1,1]$.
In addition, in relation to the above results we prove a version of Hulanicki
Theorem about inclusion of spectra for covering graphs.
