  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].

Group Theory