Finite groups with prime $p$ to the first power

Earlier D. G. Higman classified the finite groups of order n, such that n is divisible by 3 to the first power, with the assumption that the centralizer ${C_G}(X)$ of X, where X is a subgroup of order 3, is a cyclic trivial intersection set of even order 3s. In this paper the theorem is generalized to include all prime numbers greater than 3. With an additional assumption: $\vert{N_G}(X):{C_G}(X)\vert = 2$, we have proved that one of the following holds for these groups, hereafter designated as G:

(A) G is isomorphic to ${L_2}(q)$, where $q = 2ps \pm 1$;

(B) there exists a normal subgroup ${G_0}$ of odd index in G, and a normal subgroup N of ${G_0}$ of index 2 such that $G = N\langle \sigma \rangle$ where ${C_G}(X) = X \times \langle \sigma \rangle$.