  We refer to the set of the orders of elements of a finite group as its
spectrum and say that finite groups are isospectral if their spectra coincide.
In the paper we determine all finite groups isospectral to the simple groups
$S_6(q)$, $O_7(q)$, and $O_8^+(q)$. In particular, we prove that with just four
exceptions, every such a finite group is an extension of the initial simple
group by a (possibly trivial) field automorphism.
