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Let $A$ be a ring. Suppose $E$ is a right $A$-module, $F$ a left $A$-module and $G$ a $\mathbb{Z}$-module. Is there a way to define a right $A$-module structure on $\text{Hom}_\mathbb{Z}(E,G)$?

This would equivalent to defining a ring homomorphism $\phi:A^{\text{op}}\rightarrow\mathcal{E}$, where $\mathcal{E}$ is the endomorphism ring of $\text{Hom}_\mathbb{Z}(E,G)$. For $a\in A$, what should $\phi_a:\text{Hom}_\mathbb{Z}(E,G)\rightarrow \text{Hom}_\mathbb{Z}(E,G)$ be?

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You define $F$ but it doesn't appear in your question so I'm not sure you've asked the question you meant to ask. The general pattern is that $\text{Hom}$ is a contravariant functor in its first argument so it switches left and right module structures: $\text{Hom}_{\mathbb{Z}}(E, A)$ (for $A$ an abelian group; IMO it's good practice to reserve $G$ for a not-necessarily-abelian group) naturally acquires a left $A$-module structure by precomposition, and similarly $\text{Hom}_{\mathbb{Z}}(F, A)$ naturally acquires a right $A$-module structure by precomposition.

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