Authors: Akihiko Hida and Nobuaki Yagita Title: Representations of the Double Burnside Algebra and Cohomology of the Extraspecial $p$-Group Abstract: Let $E$ be the extraspecial $p$-group of order $p^3$ and exponent $p$ where $p$ is an odd prime. We determine the mod $p$ cohomology of summands in the stable splitting of $p$-completed classifying space $BE$ modulo nilpotence.