On the Schur multiplier of $p$-groups with abelianization $s$-elementary abelian

Let $p$ be an odd prime. We describe a method to compute the Schur multiplier of finite $p$-groups $G$ of nilpotency class $2$ such that $G\mathrm{/}[G,G]$ is isomorphic to direct product of copies of ${\mathbb{Z}}_{p^s}$ for $s\in \mathbb{N}$, generalizing a method of Blackburn and Evens, who treated the case $s=1$. As an application, we investigate which abelian $p$-groups can occur as the Schur multiplier of a non-abelian $p$-group. We further introduce the notions of $s$-special $p$-groups of rank $k$ generalizing the notion of special $p$-groups of rank $k$. We study the structural properties, compute the Schur multipliers of $s$-special $p$-groups of rank $1$.