Geodesic
On the product of two groups that are close to being nilpotent
Sbornik. Mathematics, Tome 38 (1981) no. 1, pp. 47-59

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The following theorem is proved. Teorem. A finite group representable as the product of two subgroups, each of which has a nilpotent subgroup of index at most $2$, is solvable. Bibliography: 19 titles. 
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L. S. Kazarin. On the product of two groups that are close to being nilpotent. Sbornik. Mathematics, Tome 38 (1981) no. 1, pp. 47-59. http://geodesic.mathdoc.fr/item/SM_1981_38_1_a3/