normal edge-transitive and 1/2-arc-transitive semi-cayley graphs

Darafsheh and assari in [normal edge-transitive cayley graphs on non-abelian groups of order 4p‎, ‎where p is a prime number‎, ‎sci‎. ‎china math‎., ‎56 (1) (2013) 213-219.] classified the connected normal edge transitive and‎ ‎12−arc-transitive cayley graph of groups of order 4p‎. ‎in this paper we continue this work by classifying the‎ ‎connected cayley graph of groups of order 2pq‎, ‎p>q are primes‎. ‎as a consequence it is proved that cay(g,s) is a‎ ‎12−arc-transitive cayley graph of order 2pq‎, ‎p>q if and only if |s| is an even integer greater than 2‎, ‎s =‎ ‎t cup t^{-1} and t subseteq { cb^ja^{i} | 0 leq i leq p‎ - ‎1}‎, ‎1 leq j leq q-1‎, ‎such that t and t^{-1} are orbits of aut(g,s) and‎ ‎begin{eqnarray*}‎ ‎g &cong& langle a‎, ‎b‎, ‎c | a^p = b^q = c^2 = e‎, ‎ac = ca‎, ‎bc = cb‎, ‎b^{-1}ab = a^r rangle‎, ‎ or‎ ‎g &cong& langle a‎, ‎b‎, ‎c | a^p = b^q = c^2 = e‎, ‎c ac = a^{-1}‎, ‎bc = cb‎, ‎b^{-1}ab = a^r rangle‎, ‎end{eqnarray*}‎ ‎where r^q equiv 1 (mod p)‎.