K₃ is a copy of unique circuit-free connected graph all of whose vertices have degree 3, called cubic tree. The group G₂² generated by x, y, t such that x²=t, y³=t²=(yt)²=1, is one of the seven finitely presented isomorphism types of sub\-groups of the full automorphism group Aut(K₃) of K₃. These seven groups act arc-transitively on the arcs of K₃ with a finite vertex stabilizer. In this paper we have found a condition on p such that the action of K₃ on the projective line over the finite field, PL(F_p), always yields the subgroups of the alternating groups of degree p+1. We have shown also that the action of K₃ on PL(F_p) is transitive.