How would I find the number of distinct homomorphisms and isomorphisms mapping the Klein four group to the Klein four group? How would I find the number of distinct homomorphisms and isomorphisms from the Klein four group to the Klein four group?
Thank you 

Here’s an approach that perhaps is more advanced, maybe even too advanced.
Your group is a two-dimensional vector space $V$ over the field $k=\Bbb F_2$ with two elements. Every homomorphism $V\to V$ is automatically a $k$-linear map, so to count these, we need only count the $2\times2$ matrices over $K$, so sixteen in number.
For automorphisms, we need to count the nonsingular matrices, or what is the same thing, the number of distinct ordered $k$-bases of $V$. To get a basis of any two-dimensional vector space, you choose first a nonzero vector (three choices, in our case), and then a vector not in the space spanned by your previous choice. This spanned space has cardinality $2$ in our case, so there are only two possible choices for the second vector. Thus six different bases in all, and six automorphisms of $V$.
