Nilpotent subgroups of self equivalences of torsion spaces

LetX be a finitep-torsion based connected nilpotent CW-complex. We give a criterion of a subgroup of ε(X), the group of self equivalences ofX, to be a nilpotent group, in terms of its action onE _*(X), whereE is a CW-spectrum, satisfying some technical conditions.