On topological groups with remainder of character κ

In [A.V. Arhangel’skii and J. van Mill, On topological groups with
a first-countable remainder, Topology Proc. 42 (2013), 157-163] it is
proved that the character of a non-locally compact topological group
with a first countable remainder doesn’t exceed w_1 and a non-locally
compact topological group of character w_1 having a compactification
whose remainder is first countable is given. We generalize these results
in the general case of an arbitrary infinite cardinal .