Let $G$ be a finite, not necessarily abelian group, and let $\text{Hom}(\mathbb{Z}^n, G)$ describe the set of group homomorphisms from $\mathbb{Z}^n\rightarrow G$.
I have been tasked with describing $\text{Hom}(\mathbb{Z}^2,G)$ as a subset of $G\times G$. I read this as trying to find a bijection between $\text{Hom}(\mathbb{Z}^2,G)$ and a subset of $G\times G$.
My first thought was to look at the isomorphism between $\text{Hom}(\mathbb{Z},G)$ and $G$; since $1$ generates $\mathbb{Z}$, we have $\phi(1)\in G$, which seems to give rise to an isomorphism between the two structures. However, after doing some reading, it seems that this is only true if $G$ is abelian.
If this kind of thinking does work, then I might be able to extrapolate using the fact that $(1,0)$ and $(0,1)$ generate $\mathbb{Z}^2$. However, I'm not convinced it does. Any suggestions/things to read would be greatly appreciated.