  For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators
$[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup
generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with
a Sylow $p$-subgroup $P$ such that any subgroup generated by a subset of
$I_G(P)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We produce
examples showing that such a result does not hold without the assumption of
$p$-solubility. Instead, we prove that if a finite group $G$ has a Sylow
$p$-subgroup $P$ such that (a) any subgroup generated by a subset of $I_G(P)$
is $r$-generated, and (b) for any $x\in I_G(P)$, any subgroup generated by a
subset of $I_G(x)$ is $r$-generated, then $[G,P]$ has $r$-bounded rank. We also
prove that if $G$ is a finite group such that for every prime $p$ dividing
$|G|$ for any Sylow $p$-subgroup $P$, any subgroup generated by a subset of
$I_G(P)$ can be generated by $r$ elements, then the derived subgroup $G'$ has
$r$-bounded rank. As an important tool in the proofs, we prove the following
result, which is also of independent interest: if a finite group $G$ admits a
group of coprime automorphisms $A$ such that any subgroup generated by a subset
of $I_G(A)$ is $r$-generated, then the rank of $[G,A]$ is $r$-bounded.
