  Motivated by recent advances in Donaldson-Thomas theory, four-dimensional
$\mathcal{N}=4$ string-string duality is examined in a reduced rank theory on a
less studied BPS sector. In particular we identify candidate partition
functions of "untwisted" quarter-BPS dyons in the heterotic $\mathbb{Z}_2$ CHL
model by studying the associated chiral genus two partition function, based on
the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields
meromorphic Siegel modular forms for the Iwahori subgroup $B(2) \subset
\text{Sp}_4 (\mathbb{Z}) $ which generate BPS indices for dyons with untwisted
sector electric charge, in contrast to twisted sector dyons counted by a
multiplicative lift of twisted-twining elliptic genera known from Mathieu
moonshine. The new partition functions are shown to satisfy the expected
constraints coming from wall-crossing and S-duality symmetry as well as the
black hole entropy based on the Gauss-Bonnet term in the effective action. In
these aspects our analysis confirms and extends work of Banerjee, Sen and
Srivastava, which only addressed a subset of the untwisted sector dyons
considered here. Our results are also compared with recently conjectured
formulae of Bryan and Oberdieck for the partition functions of primitive DT
invariants of the CHL orbifold $X=( \text{K3} \times T^2 )/ \mathbb{Z}_2$, as
suggested by string duality with type IIA theory on $X$.
