  We study ${\cal N}=2$ compactifications of $E_8\times E_8$ heterotic string
theory on orbifolds of $K3 \times T^2$ by $g'$ which acts as an $\mathbb{Z}_N$
automorphism of $K3$ together with a$1/N$ shift on a circle of $T^2$. The
orbifold action $g'$ corresponds to the $26$ conjugacy classes of the Mathieu
group $M_{24}$. We show that for the standard embedding the new supersymmetric
index for these compactifications can always be decomposed into the elliptic
genus of $K3$ twisted by $g'$. The difference in one-loop corrections to the
gauge couplings are captured by automorphic forms obtained by the theta lifts
of the elliptic genus of $K3$ twisted by $g'$. We work out in detail the case
for which $g'$ belongs to the equivalence class $2B$. We then investigate all
the non-standard embeddings for$K3$ realized as a $T^4/\mathbb{Z}_\nu$ orbifold
with $\nu = 2, 4$ and $g'$ the $2A$ involution. We show that for non-standard
embeddings the new supersymmetric index as well as the difference in one-loop
corrections to the gauge couplings are completely characterized by the
instanton numbers of the embeddings together with the difference in number of
hypermultiplets and vector multiplets in the spectrum.
