Enumerating isoclinism classes of semi-extraspecial groups

We enumerate the number of isoclinism classes of semi-extraspecial $p$-groups with derived subgroup of order $p^2$. To do this, we enumerate $$\text{GL}$(2, p)$-orbitsofsetsofirreducible,monicpolynomialsin$$\mathbb{F}$_p[x]$. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.