On noninner automorphisms of finite p-groups that fix the Frattini subgroup elementwise

Let [Formula: see text] be a finite [Formula: see text]-group and let [Formula: see text]. In this paper we show that if [Formula: see text] lies in the second center [Formula: see text] of [Formula: see text], then [Formula: see text] admits a noninner automorphism of order [Formula: see text], when [Formula: see text] is an odd prime, and order [Formula: see text] or [Formula: see text], when [Formula: see text]. Moreover, the automorphism can be chosen so that it induces the identity on the Frattini subgroup [Formula: see text]. When [Formula: see text], this reduces the verification of the well-known conjecture that states every finite nonabelian [Formula: see text]-group [Formula: see text] admits a noninner automorphism of order [Formula: see text] to the case in which [Formula: see text] where [Formula: see text]. In addition, it follows that if [Formula: see text] is a finite nonabelian [Formula: see text]-group, [Formula: see text], such that [Formula: see text] is a cohomologically trivial [Formula: see text]-module, then [Formula: see text] satisfies the above mentioned condition, and as a consequence we show that the order of [Formula: see text] is at least [Formula: see text].