Class-preserving automorphisms of some finite $p$-groups

Let $G$ be a finite $p$-group of order $p^5$, where $p$ is a prime. We give necessary and sufficient conditions on $G$ such that $G$ has a non-inner class-preserving automorphism. As a consequence, we give short and alternate proofs of results of section 5 of Yadav [Proc. Indian Acad. Sci. (Math. Sci.) 118 (2008), 1-11] and Theorem 4.2 of Kalra and Gumber [Indian J. Pure Appl. Math. 44 (2013), 711-725].

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