P. Hall's covering group and embeddings of countable cc-groups

Let A be an elementary abelian group of order at least p ³ acting on a finite p′-group G that is soluble with derived length d. Assume that γ _c (C _G (a)) has exponent dividing m for any a ∈ A ^#. It is proved that there exist {p, d, c, m}-bounded numbers c ₁ and m ₁ such that γ _{c 1} (G) has exponent dividing m ₁.