Finite groups of automorphisms of Enriques surfaces and the Mathieu group $M_{12}$

An action of a group $G$ on an Enriques surface $S$ is called Mathieu if it acts on $H^0(2K_S)$ trivially and every element of order 2, 4 has Lefschetz number 4. A finite group $G$ has a Mathieu action on some Enriques surface if and only if it is isomorphic to a subgroup of the symmetric group ${\mathfrak{S}}_6$ of degree 6 and the order $\mathrm{\mid }G\mathrm{\mid }$ is not divisible by $2^4$. Explicit Mathieu actions of the three groups ${\mathfrak{S}}_5,N_{72}$ and ${\mathfrak{A}}_6$, together with non-Mathieu one of $H_{192}$, on polarized Enriques surfaces of degree 30, 18, 10 and 6, respectively, are constructed without Torelli type theorem to prove the if part.