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 {\em homomorphism reducible} to $L$ iff there is a continuous group homomorphism $\phi :G\to G$ such that $H = \varphi^{-1} (L)$\mathrm{.}Wepreviouslyshowedthatthereisa$K_\sigma$subgroup,$L$,ofthecountablepowerofanylocallycompactPolishgroup,$G$, 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.