Homomorphism reductions on Polish groups

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and (H,L subseteq G) are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism (varphi : G rightarrow G) such that (H = varphi ^{-1} (L)). We previously showed that there is a (K_sigma ) subgroup L of the countable power of any locally compact Polish group G such that every (K_sigma ) subgroup of (G^omega ) is homomorphism reducible to L. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.