Okay, abhishekkgp pointed out to me that it is a lot harder than I first thought to find a homomorphism of a group with its subgroups. There is a way, but it's more reminiscent of a trivial homomorphism and I was hoping to avoid that. For example, take D4. We can define an endomorphism from D4 to its subgroup C4:
(I'll define the various reflection planes if anyone cares to know them.) This is a homomorphism, but not a very interesting one.