On ηα-groups and fields

Let κ:=ℵα. The following are known: two ηα-sets of power κ are isomorphic. Let α>0. Two ordered divisible Abelian groups that are ηα-sets of power κ are isomorphic, two real closed fields that are ηα-sets of power κ are isomorphic. The following is shown: (1) there exist 2κ nonisomorphic ordered Abelian groups (respectively ordered fields) that are ηα-sets of power κ; (2) there exist 2κ nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power κ all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are ηα-sets having the same order type.