Automorphism of finite 2-group of Mersenne prime order acts nontrivially on Omega-1

Statement
Suppose $$P$$ is a finite $$2$$-group, and $$\sigma$$ is a non-identity automorphism of $$P$$ such that the order of $$\sigma$$ is a Mersenne prime greater than $$3$$. Then, $$\sigma$$ acts nontrivially on the subgroup $$\Omega_1(P)$$ (the first omega subgroup).

Related facts

 * Omega-1 of odd-order p-group is coprime automorphism-faithful