Automorphisms of a finite $p$-group with cyclic Frattini subgroup

Let $G$ be a group and $Aut^{\Phi}(G)$ denote the group of all automorphisms of $G$ centralizing $G/\Phi(G)$ elementwise‎. ‎In this paper‎, ‎we characterize the finite $p$-groups $G$ with cyclic Frattini subgroup for which $|Aut^{\Phi}(G):Inn(G)|=p$‎.

Keywords:

‎‎Automorphism group‎ , ‎Finite $p$-group‎ , ‎Frattini subgroup‎