On automorphism groups of Riemann double covers of Klein surfaces

If G is a group of automorphisms of a compact Klein surface X, then the direct product G×C2 is a group of automorphisms of the Riemann double cover X+ of X. In this paper we analyse the relationship between G and the full groups of automorphisms Aut(X) and Aut(X+) of X and X+ respectively, in the special case where the group G is uniformised by a non-Euclidean crystallographic group with quadrangular signature (2,2,2,n). There is a difference in what happens between bordered surfaces and unbordered non-orientable surfaces, and so we consider those cases separately (including the special situation for n=4 in the unbordered case).