Morita equivalence between $\mathbb{C}[G]$ and $\mathbb{C}[H]$?

Question

What we can say about two groups G and H when their group rings, $\mathbb{C}[G]$ and $\mathbb{C}[H]$, are Morita equivelent?

Answer 1

I will assume that $G$ and $H$ are finite groups, since I don't know the theory otherwise. Then it just says that $G$ and $H$ have the same number of conjugacy classes.

Indeed, $\mathbb{C}[G]$ is Morita-equivalent to $\mathbb{C}^m$ where $m$ is the number of conjugacy classes of $G$, and $\mathbb{C}^m$ is Morita-equivalent to $\mathbb{C}^n$ iff $m=n$.