A theorem about coprime action

It is well known that if an elementary abelian p-group P acts on a p′-group Q and Q=[Q,P], then Q=〈[CQ(A),P]|A⩽P of index p〉. Does a similar statement hold for CQ(P)? Under further assumptions, the answer is yes. Goldschmidt proves theorems of this flavour in [D.M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972) 341–351. [1], ; D.M. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975) 475–489. [2]] and uses them to construct signalizer functors. For the same reason we prove a result of this type, under the assumption that Q is soluble.