@TheMaskedAvenger I am not quite sure what you mean by products that operate on groups (usually, the groups are the ones that operate on things, and it just sometimes happens that those things are themselves groups)

Your use of the word product reminded me of the various ways of constructing groups from a tuple of (copies of) groups. I was musing on the various constructions and wondering how "invertible" they are, and also if there is a nice characterization of products that are easily deconstructed.

Well, the various products (at least the ones I am familiar with) come in sort of a hierarchy, ranging from most general to least (knit product, semidirect product, direct product) (and wreath product is a special case of a semidirect product)

as to how invertible they are, as far as I recall, not even the direct product is particularly invertible unless one assumes something extra about the group(s)