  Let $G$ be a discrete group, let $p\ge1$, and let $\omega$ be a weight on
$G$. Using the approach from [9], we provide sufficient conditions on a weight
$\omega$ for $\ell^p(G,\omega)$ to be a Banach algebra admitting a
norm-controlled inversion in the reduced C$^*$-algebra of $G$, namely
$C^*_r(G)$. We show that our results can be applied to various cases including
locally finite groups as well as finitely generated groups of polynomial or
intermediate growth and a natural class of weights on them. These weights are
of the form of polynomial or certain subexponential functions. We also consider
the non-discrete case and study the existence of norm-controlled inversion in
$B(L^2(G))$ for some related convolution algebras.
