The Local Oort conjecture (now a theorem of Obus-Wewers and Pop) states that if k is an algebraically closed field of characteristic p, then any G-Galois extension k[[z]]/k[[t]] with G cyclic lifts to characteristic zero. That is, it lifts to a G-Galois extension R[[Z]]/R[[T]], where R is a characteristic zero DVR with residue field k. The same is true if G is A_4 and p = 2 (Bouw), or if G is dihedral of order 2p (Pagot, Bouw-Wewers). In the other direction, it follows from work of Chinburg-Guralnick-Harbater and Brewis-Wewers that if G is a so-called "local Oort group" for p (i.e., all G-Galois extensions lift from characteristic p to characteristic zero), then G is either cyclic, A_4 for p = 2, or dihedral of order 2p^n. We take the first steps toward showing that dihedral groups of order 2p^n are in fact local Oort groups for odd p and arbitrary n. In particular, we give the first examples of liftable extensions when p is odd and n > 1.