On the nilpotency of the solvable radical of a finite group isospectral to a simple group

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.
Comment: arXiv admin note: text overlap with arXiv:1806.07045