On almost recognizability by spectrum of simple classical groups

The set of element orders of a finite group G is called the {\em spectrum}ý. ýGroups with coinciding spectra are said to be {\em isospectral}ý. ýIt is known that if G has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphicý ýgroups isospectral to Gý. ýThe situation is quite different if G is a nonabelain simple groupý. ýRecently it was proved that if L is a simple classical group of dimension at least 62 and G is a finite groupý ýisospectral to Lý, ýthen up to isomorphism L≤G≤\AutLý. ýWe show that the assertion remains trueý ýif 62 is replaced by 38ý.