smc

The first three listed operations, applied to even integers, all yield even integers trivially. As for the fourth operation (division) and fifth operation (average of pair), we can find pairs of even numbers that are mapped to odd integers, such as $\frac{2}{2}=1$, and $\frac{0+2}{2}=1$. Therefore, the operations that map even integers to even integers only are operations $1,2,3$, so our answer is $\boxed{1,2,3}$ and we are done.