Characterization of the linear groups PSL(2,11) and PSL (2,13)

We prove the nonsimplicity of a finite group containing an involution τ such that the quotient group C(τ)/{τ} the Frobenius group with an additional factor of odd prime order acting transitively on the nonunit elements of the kernel. Based on this we obtain a characterization of the linear groups PSL(2,11) and PSL(2,13).