noninner automorphisms of finite $p$-groups leaving the center elementwise fixed

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. let $g$ be a finite nonabelian $p$-group. it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic, or $g/z(g)$ is powerful, then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattini subgroup $phi(g)$ of $g$ elementwise fixed. in this note, we prove that the latter noninner automorphism can be chosen so that it leaves $z(g)$ elementwise fixed.