On finite groups isospectral to simple linear and unitary groups

Let $L$ be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic $p$. We deal with the class of finite groups isospectral to $L$. It is known that a group of this class has a unique nonabelian composition factor. We prove that if $L\ne U_4(2),U_5(2)$ then this factor is isomorphic to either $L$ or a group of Lie type over a field of characteristic different from $p$.