Pentavalent vertex-transitive diameter two graphs

We classify the family of pentavalent vertex-transitive graphs Γ with diameter 2. Suppose that the automorphism group of Γ is transitive on the set of ordered distance 2 vertex pairs. Then we show that either Γ is distance-transitive or Γ is one of \(\overline {{C_8}} ,{\kern 1pt} {K_5}\square {K_2},{\kern 1pt} {C_5}\left {{K_2}} \right],{\kern 1pt} \overline {2{C_4}} ,{\kern 1pt} or{\kern 1pt} {K_3}\square {K_4}\).