On the conjecture of non-inner automorphisms of finite $p$-groups with a non-trivial abelian direct factor

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that the conjecture is true when a finite non-abelian $p$-group $G$ has a non-trivial abelian direct factor. Moreover, we prove that the non-inner automorphism is central and fixes $\mathrm{\Phi }(G)$ elementwise. As a consequence, we prove that every group which is not purely non-abelian has a non-inner central automorphism of order $p$ which fixes $\mathrm{\Phi }(G)$ elementwise.