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We study ${\cal N}=2$ compactifications of $E_8\times E_8$ heterotic stringtheory 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 $T^2$. Theorbifold action $g'$ corresponds to the $26$ conjugacy classes of the Mathieugroup $M_{24}$. We show that for the standard embedding the new supersymmetricindex for these compactifications can always be decomposed into the ellipticgenus of $K3$ twisted by $g'$. The difference in one-loop corrections to thegauge couplings are captured by automorphic forms obtained by the theta liftsof the elliptic genus of $K3$ twisted by $g'$. We work out in detail the casefor which $g'$ belongs to the equivalence class $2B$. We then investigate allthe non-standard embeddings for$K3$ realized as a $T^4/\mathbb{Z}_\nu$ orbifoldwith $\nu = 2, 4$ and $g'$ the $2A$ involution. We show that for non-standardembeddings the new supersymmetric index as well as the difference in one-loopcorrections to the gauge couplings are completely characterized by theinstanton numbers of the embeddings together with the difference in number ofhypermultiplets and vector multiplets in the spectrum.

${\cal N}=2$ heterotic string compactifications on orbifolds of $K3\times T^2$