In [H. Huang, N.L. Johnson: Semifield planes of order 82, Discrete Math., 80 (1990)], the authors exhibited seven sporadic semifields of order 2^6, with left nucleus F_{2^3} and center F_2. Following the notation of that paper, these examples are referred as the Huang–Johnson semifields of type II, III, IV , V , V I, V II and V III. In [N. L. Johnson, V. Jha, M. Biliotti: Handbook of Finite Translation Planes, Pure and Applied Mathematics, Taylor Books, 2007], the question whether these semifields are contained in larger families, rather then sporadic, is posed. In this paper, we first prove that the Huang–Johnson semifield of type V I is isotopic to a cyclic semifield, whereas those of types V II and V III belong to infinite families recently constructed in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti: Semifields of order q^6 with left nucleus F_{q^3} and center F_q, Finite Fields Appl., 14 (2008)] and [G.L. Ebert, G. Marino, O. Polverino, R. Trombetti: Infinite families of new semifields, Combinatorica, 6 (2009)]. Then, Huang–Johnson semifields of type II and III are extended to new infinite families of semifields of order q6, existing for every prime power q.