On al most recognizability by spectrum of simple classical groups

The set of element orders of a finite group G is called the spectrum. Groups with coinciding spectra are said to be isospectral. It is known that if G has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphic groups isospectral to G. The situation is quite different if G is a nonabelain simple group. Recently it was proved that if L is a simple classical group of dimension at least 62 and G is a finite group isospectral to L, then up to isomorphism L ≤ G ≤ Aut L. We show that the assertion remains true if 62 is replaced by 38.