Beyl-Tappe [4] say on p. 119:
5.9 REMARK. The Schur multiplicators of all finite simple groups have been found, often by exhibiting a universal perfect cover (representation group). For the results see the summary by GRIESS [2] and follow the references given there.
[1] Griess, Robert L. jun., Schur multipliers of the known finite simple groups, Bull. Am. Math. Soc. 78, 68-71 (1972). ZBL0263.20008.
[2] Griess, Robert L. jun., Schur multipliers of the known finite simple groups. II, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 279-282 (1980). ZBL0448.20014.
[3] Griess, Robert L. jun., Schur multipliers of the known finite simple groups. III, Proc. Rutgers group theory year, 1983/84, 69-80 (1984). ZBL0648.20017.
[4] Beyl, F. Rudolf; Tappe, Jürgen, Group extensions, representations, and the Schur multiplicator, Lecture Notes in Mathematics. 958. Berlin-Heidelberg-New York: Springer-Verlag. IV, 278 p. DM 33.50 {$} 13.50 (1982). ZBL0544.20001.