  We study three properties of $1/4$ BPS dyons at small charges in string
compactifications which preserve ${\cal N}=4$ supersymmetry. We evaluate the
non-trivial constant present in the one loop statistical entropy for ${\cal
N}=4$ compactifications of type IIB theory on $K3\times T^2$ orbifolded by an
order $\mathbb{Z}_N$ freely acting orbifold $g'$ including all CHL
compactifications. This constant is trivial for the un-orbifolded model but we
show that it contributes crucially to the entropy of low charge dyons in all
the orbifold models. We then show that the meromorphic Jacobi form which
captures the degeneracy of $1/4$ BPS states for the first two non-trivial
magnetic charges can be decomposed into an Appell-Lerch sum and a mock Jacobi
form transforming under $\Gamma_0(N)$. This generalizes the earlier observation
of Dabholkar-Murthy-Zagier to the orbifold models. Finally we study the sign of
the Fourier coefficients of the inverse Siegel modular form which counts the
index of $1/4$ BPS dyons in ${\cal N}=4$ models obtained by freely acting
$\mathbb{Z}_2$ and $\mathbb{Z}_3$ orbifolds of type II theory compactified on
$T^6$. We show that sign of the index for sufficiently low charges and ensuring
that it counts single centered black holes, violates the positivity conjecture
of Sen which indicates that these states posses non-trivial hair.
