On orders of elements of finite almost simple groups with linear or unitary socle

We say that a finite almost simple G with socle S is admissible (with respect to the spectrum) if G and S have the same sets of orders of elements. Let L be a finite simple linear or unitary group of dimension at least three over a field of odd characteristic. We describe admissible almost simple groups with socle L. Also we calculate the orders of elements of the coset Lτ, where τ is the inverse-transpose automorphism of L.