A group is called 2-genetic if each normal subgroup of the group can be generated by two elements. Let G be a non-abelian 2-genetic group of order p^n for an odd prime p and a positive integer n. In this paper, we investigate connected Cayley digraphs Cay(G, S), and determine their full automorphism groups when Aut(G, S) = {α\in Aut(G) | S^α = S} is a p′-group. With the result, we give the first known half-arc-transitive non-normal Cayley graphs of order an odd prime-power.