Finite groups and the fixed points of coprime automorphisms

Let $p$ be a prime, and let $G$ be a finite $p'$-group acted on by an elementary abelian $p$-group $A$. The following results are proved:

1. If $\vert A\vert\ge p^3$ and $C_G(a)$ is nilpotent of class at most $c$ for any $a\in A^\char93$, then the group $G$ is nilpotent of $\{c,p\}$-bounded class.

2. If $\vert A\vert\ge p^4$ and $C_G(a)'$ is nilpotent of class at most $c$ for any $a\in A^\char93$, then the derived group $G'$is nilpotent of $\{c,p\}$-bounded class.