  We enumerate the number of isoclinism classes of semi-extraspecial $p$-groups
with derived subgroup of order $p^2$. To do this, we enumerate $\text{GL}(2,
p)$-orbits of sets of irreducible, monic polynomials in $\mathbb{F}_p[x]$.
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.
