Monochrome symmetric subsets of colored groups

In (Electron. J. Combin. 10 (2003); http://www.combinatorics.org/volume-10/s/v1oi1r28.html), the first author (Yuliya Gryshko) asked three questions. Is it true that every infinite group admitting a 2-coloring without infinite monochromatic symmetric subsets is either almost cyclic (i.e., have a finite index subgroup which is cyclic infinite) or countable locally finite? Does every infinite group G include a monochromatic symmetric subset of any cardinal