A generalization of some of Folkman’s constructions (see (1967) J. Comb. Theory, 3, 215–232) of the so-called semisymmetric graphs, that is regular graphs which are edge- but not vertex-transitive, was given by Marui and Potonik (2001, Europ. J. Combinatorics,22, 333–349) together with a natural connection between graphs admitting 1 / 2-arc-transitive group actions and certain graphs admitting semisymmetric group actions. This connection is studied in more detail in this paper. Among others, a sufficient condition for the semisymmetry of the so-called generalized Folkman graphs arising from certain graphs admitting a 1 / 2-arc-transitive group action is given. Furthermore, the concepts of alter-sequence and alter-exponent is introduced and studied in great detail and then used to study the interplay of three classes of graphs: cubic graphs admitting a one-regular group action, the corresponding line graphs which admit a 1 / 2-arc-transitive action of the same group and the associated generalized Folkman graphs. At the end an open problem is posed, suggesting an in-depth analysis of the structure of tetravalent 1 / 2-arc-transitive graphs with alter-exponent 2.