There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature.
Burgoyne & Fong (1966) incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction Burgoyne & Fong (1968) incorrectly claimed that it has order 6.
Mazet (1979) showed that the Schur multiplier is in fact cyclic of order 12.
M22 has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 24:A6, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A7.
The triple cover 3.M22 has a 6-dimensional faithful representation over the field with 4 elements.