Let $p$ and $q$ be distinct primes. Suppose we have a group $G = P \rtimes Q,$ where $P$ is a non-abelian $p$-group and $Q$ is an abelian $q$-group such that it is a subgroup of ${\rm Aut}(P).$
Then we know that $P = C_P(Q)[P,Q].$
But from here I am unable to conclude anything about $P'?$ Any information or reference would be highly appreciated! Thanks