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Apr 3, 2018 at 22:06 comment added R W I understand the question - still in order to formulate a rigorous conjecture one has to make more effort. However the convolution formula I give leaves little hope for "splitting" the convolution on the original group into the convolution on the quotient group and a certain convolution like operation on the fibers. What if you look first at what happens in the simplest situation when the group $G$ is just the product of the quotient group and the kernel?
Apr 3, 2018 at 21:18 comment added Alex M. Does the edit (last paragraph) that I have performed on my question clarify what I am looking for?
Apr 3, 2018 at 20:34 history answered R W CC BY-SA 3.0