Twisted conjugacy in Houghton's groups

For a fixed n∈{2,3,…}, the Houghton group Hn consists of bijections of Xn={1,…,n}×N that are 'eventually translations' of each copy of N. The Houghton groups have been shown to have solvable conjugacy problem. In general, solvable conjugacy problem does not imply that all finite extensions and finite index subgroups have solvable conjugacy problem. Our main theorem is that a stronger result holds: for any n∈{2,3,…} and any group G commensurable to Hn, G has solvable conjugacy problem.