The Schur multiplier of the Mathieu group M₂₂ is cyclic of order 12. The history of its calculating is rather dramatical since the result was not got right at once. I will discuss an alternative way of calculating the Schur multiplier of M₂₂. As an intermediate result it is reproved that the Schur multiplier of L₃(4) (which is the direct product of the cyclic group of order 4 and the cyclic group of order 12). We consider this results in context of a larger project of revisiting the calculating the Schur multipliers of all the known finite simple groups.