  In this paper we study congruence properties of the representations
$U_\alpha:=U^{PSL(2,\mathbb{Z})}_{\chi_\alpha}$ of the projective modular group
${\rm PSL}(2,\mathbb{Z})$ induced from a family $\chi_\alpha$ of characters for
the Hecke congruence subgroup $\Gamma_0(4)$ basically introduced by A. Selberg.
Interest in the representations $U_\alpha$ stems from their appearance in the
transfer operator approach to Selberg's zeta function for this Fuchsian group
and character $\chi_\alpha$. Hence the location of the nontrivial zeros of this
function and therefore also the spectral properties of the corresponding
automorphic Laplace-Beltrami operator $\Delta_{\Gamma,\chi_\alpha}$ are closely
related to their congruence properties. Even if as expected these properties of
$U_\alpha$ are easily shown to be equivalent to the ones well known for the
characters $\chi_\alpha$, surprisingly, both the congruence and the
noncongruence groups determined by their kernels are quite different: those
determined by $\chi_\alpha$ are character groups of type I of the group
$\Gamma_0(4)$, whereas those determined by $U_\alpha$ are such character groups
of $\Gamma(4)$. Furthermore, contrary to infinitely many of the groups $\ker
\chi_\alpha $, whose noncongruence properties follow simply from Zograf's
geometric method together with Selberg's lower bound for the lowest
nonvanishing eigenvalue of the automorphic Laplacian, such arguments do not
apply to the groups $\ker U_\alpha$, the reason being, that they can have
arbitrary genus $g\geq 0$, contrary to the groups $\ker \chi_\alpha$, which all
have genus $g=0$.
