C₄ is the cyclic group of order 4. {0,1,2,3} with addition modulo 4 as the operation.

A single element, either 1 or 3 will generate it. I pick 1. To specify an endomorphism, f, chose f(1). f(0) has to be 0 anyway and f(2) and f(3) follow from f(2)=f(1+1)=f(1)+f(1) and f(3)=f(1+1+1)=f(1)+f(1)+f(1).

Looks to be like I have four choices.

f(1)=0

This is the boring one, everything is mapped to zero. Call it null

f(1)=1

This is the indentity map. Call it identity

f(1)=3

This fixes f(2) at 3+3=2, and f(3) = 3+3+3 =9=2×4+1=1. 0 and 2 don't move, but 1 and 3 are swapped. Call this swap

f(1)=2

Up till now, either the kernel or the image has been trivial. Mapping 1 to 2 takes 2 to 4=0, so the kernel is {0,2}. f(3)=f(1+1+1)=2+2+2=2. The image is also {0,2}. Since this is less than the whole group I'm going to call this one shrink

Now that I've got the endomorphisms of an Abelian group I can make a ring, using compostion for multiplication and point-wise addition of functions for addition. I think I'll leave my endomorphisms to hang for a day or two before I try cooking this particular dish