  This work is devoted to the study of Bessel and Riesz systems of the type
$\big\{L_{\gamma}\mathsf{f}\big\}_{\gamma\in \Gamma}$ obtained from the action
of the left regular representation $L_{\gamma}$ of a discrete non abelian group
$\Gamma$ which is a semidirect product, on a function $\mathsf{f}\in
\ell^2(\Gamma)$. The main features about these systems can be conveniently
studied by means of a simple matrix-valued function $\mathbf{F}(\xi)$. These
systems allow to derive sampling results in principal $\Gamma$-invariant
spaces, i.e., spaces obtained from the action of the group $\Gamma$ on a
element of a Hilbert space. Since the systems
$\big\{L_{\gamma}\mathsf{f}\big\}_{\gamma\in \Gamma}$ are closely related to
convolution operators, a connection with $C^*$-algebras is also established.
