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Randy, I am not sure what you mean by "family group" but a hybrid is an offspring between two different species, and a cross is between the same species but different varieties. Take a dog for instance, if you mate a poodle with a retriever (both the same species, Canis lupus familiaris) it is not a hybrid, it is still the same species as any other dog. Now take a horse and mate it with a donkey (same genus, Equus, but different species) and you get a mule which is a hybrid and normally not fertile, although every once in a while they do get a fertile mule, but it is very rare.
If by group you mean genus, they are still hybrids but they are put in the same genus because of a very close relationship, in fact animals like a golden pheasant and Lady Amherst pheasant are often called sister species because of the close genetic relationship, but even closely related sister species may not produce fertile young, it depends on how closely the genetics match.
EDIT, sorry I didn't read all of the posts because this was a reply to a different thread that I was linked to and I see others have given much more detailed responses, this is just a simplified version of some of the deffinitions.