  We consider $AD$-type orbifolds of the triplet vertex algebras
$\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex
algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$
and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral
groups, respectively. A combinatorial algorithm for classification of
irreducible $\mathcal{W}(p)^\Gamma$-modules is developed, which relies on a
family of constant term identities and properties of certain polynomials based
on constant terms. All these properties can be checked for small values of $m$
and $p$ with a computer software. As a result, we argue that if certain
constant term properties hold, the irreducible modules constructed in [Commun.
Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J.
Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of
irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This
paper is a continuation of our previous work on the $ADE$ subalgebras of the
triplet vertex algebra $\mathcal{W}(p)$.
