Another application (known to the OP, but interesting enough to describe clearly) is a result of Burnside (1905) classifying for which powers b = b(p, a, q) there is a group of order paqb with no non-identity normal p-subgroup: the classification is based simply on the orders of the automorphism groups of the p-subgroups. Burnside had an error in his analysis of the associated arithmetical condition that was corrected in Coates–Dwan–Rose (1976). Burnside's result was generalized in Glauberman (1975) and Bialostocki (1975, 1987). Many of these and further results are based on analyzing nilpotent p′-subgroups of GL(n, p), resting ultimately on the fact that that every p′-subgroup of the automorphism group of a p-group of rank n is isomorphic (including in some sense, its action) to a subgroup of GL(n, p).
Burnside, W.On groups of order pαqβLond. M. S. Proc. (2) 1, 388-392 (1904).JFM35.0162.01DOI:10.1112/plms/s2-1.1.388
Burnside, W.On groups of order pαqβ (second paper).Lond. M. S. Proc. (ser 2) 2, (1905) 432-437.JFM36.0198.02DOI:10.1112/plms/s2-2.1.432
Glauberman, G.On Burnside's other paqb theorem.Pacific J. Math. 56 (1975), no. 2, 469–476.MR412269URL:euclid.pjm/1102906371
Bialostocki, Arie.On products of two nilpotent subgroups of a finite group.Israel J. Math. 20 (1975), no. 2, 178–188.MR407148DOI:10.1007/BF02757885
Coates, Martin; Dwan, Michael; Rose, John S.A note on Burnside's other pαqβ theorem.J. London Math. Soc. (2) 14 (1976), no. 1, 160–166.MR419594DOI:10.1112/jlms/s2-14.1.160
Bialostocki, Arie.On the other pαqβ theorem of Burnside.Groups–St. Andrews 1985.Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 1, 41–49.MR879428DOI:10.1017/S0013091500017946
O'Brien, E. A.(5-ANUM)