Let $\{G_\alpha\}$ be a collection of groups and $G$ be a group.
Can we interpret $(\oplus_\alpha G_\alpha)\ast G$ in terms of $G_\alpha \ast G$ and $G_\alpha \oplus G$?
Edit : At first glance, I thought that $(\mathbb{Z}\oplus\mathbb{Z})\ast\mathbb{Z}=\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z}$.
But, I think that it is not true.