  Let $G$ be a finite group acting on $\mathbb{C}^N$. We study the problem of
identifyng the class in $\mathbb{C}^N / G$ of a given signal: this encompasses
several types of problems in signal processing. Some instances include certain
generalizations of phase retrieval, image recognition, the analysis of
textures, etc. In our previous work \cite{prev}, based on an algebraic
approach, we constructed a Lipschitz translation invariant transform -- the
case when $G$ is cyclic. Here, we extend our results to include all finite
Abelian groups. Moreover, we show the existence of a new transform that avoids
computing high powers of the moduli of the signal entries--which can be
computationally taxing. The new transform does not enjoy the algebraic
structure imposed in our earlier work and is thus more flexible. Other (even
lower) dimensional representations are explored, however they only provide an
almost everywhere (actually generic) recovery which is not a significant
drawback for applications. Our constructions are locally robust and provide
alternatives to other statistical and neural networks based methods.
