Let $P$ be a finite $p$-group ($p$ prime) and $C$ a finite cyclic group of order coprime to $p$.

What can we say about the representation ring of a semi-direct product $R_{\mathbb Q}(P \rtimes C )$?

I'm looking for something similar to the case of a direct product, where we have $R_{\mathbb Q}(P \times C )\cong R_{\mathbb Q}(P)\times R_{\mathbb Q}(C)$.