Rank type conditions on commutators in finite groups

For a subgroup S of a group G, let IG(S) denote the set of commutators [g,s]=g⁻¹gˢ, where g∈ G and s∈ S, so that $[G,S]$ is the subgroup generated by IG(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 IG(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 IG(P) is r-generated, and (b) for any x∈ IG(P), any subgroup generated by a subset of IG(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 IG(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 IG(A) is r-generated, then the rank of $[G,A]$ is r-bounded.