Perhaps this is a trivial question, but I have no idea how to justify it.

Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and multiplication $(+, \ast)$ say satisfying the condition that $(R, +)$ is isomorphic to $G_1$ and the group of units of $R$, say $U(R)$, satisfy $(U(R), \ast) \cong G_2$.

For instance, the existence of finite fields of order $p^n$ for a prime $p$ implies that the pair $(G_1, G_2)$ with $G_1$ being the group $(C_p)^n = C_p \times \cdots \times C_p$ and $G_2$ being the cyclic group of order $p^n - 1$ is a ring-compatible pair.

Are all pairs of groups $(G_1, G_2)$ with $G_1$ abelian and if $G_1, G_2$ are finite groups, $|G_1| > |G_2$, ring compatible?

I apologize in advance if this question is trivial.

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