In the author's previous paper [Part I, Algebra Logika 19, 91-102 (1980; Zbl 0475.20017)] it was proved that if G is a finite group with a self- normalizing subgroup \(\) of order 6, then G is a solvable group of 3- length 1, or \(x^ 2\not\in G'.\) The main result of this paper is Theorem. Let G be a finite group with a self-normalizing subgroup \(\) of order 6, \(t=x^ 3\), \(f=x^ 2\). Then \(tO_ 2(G)\in Z^*(G/O_ 2(G)),\) or \(G=(F(G)\times E(G)),\) where \(E(G)\cong^ 2G_ 2(3)\) and f acts fixed-point-freely on F(G). Moreover there is a series of interesting propositions in this paper. Proposition 1. Let t be an involution which lies in only one Sylow 2- subgroup T of the finite group G, z is an involution from \(C_ T(\Omega_ 1(C_ T(t))).\) If \(z\in O_ 2(C(u))\) for every involution u from \(C_ T(t)\), then \(z\in O_ 2(G)\), or z and t are conjugate and \(zO_ 2(G)\in Z^*(G/O_ 2(G)),\) or \(G/O_ 2(G)\) is a covering group of \(L_ 2(q)\), Sz(q), \(U_ 3(q)\), where q is even. Proposition 2. Let a finite group G contain an involution t, such that N(X) is a 2-composed group \((N(X)=N=N_ 2\times N_{2'})\) for every 2- subgroup X from G containing t. Then \(tO_ 2(G)\in Z^*(G/O_ 2(G)).\)