  We prove a result on approximate recovery, with high probability, of
subgroups of a finite nonabelian group $\Gamma$ from their random
perturbations. We use this for ad-hoc sequences of $\Gamma_n$ while passing to
the continuum limit, in order to obtain asymptotic almost sure recovery for
rational lcsc nilpotent Lie groups. By comparison to limit theorems for groups
of polynomial growth, it turns out that this setting is the natural general
setting for recovery results, under polynomial growth assumptions on the
$\Gamma_n$. This approach makes effective the convergence rate in previous
Fourier recovery theorems in Euclidean space, and extends them to the
nonabelian setting. A series of interesting further directions are highlighted
by this approach.
