As a post grad student, I have come across many results where a function with certain properties(eg-homomorphism) on a smaller algebraic structure is extended to a larger one. For example,
extending a multiplicative character from a subgroup of a finite Abelian group to the entire group.
extending monomorphisms to automorphisms in case of normal extension.
extending to basis of a finite dimensional vector space.
I would like to have a list of such extension theorems and also learn about some general ideas on how we can extend a function. Of course, there are many such in analysis and topology as well but let us stick to algebra for now. I hope this is the right place to ask. If not, please guide me.