say you have two subgroups $\displaystyle G\text{ and }H$ and an order preserving group homomorphism $\displaystyle \eta:G\rightarrow H$ which maps the identity of $\displaystyle G$ to the identity of $\displaystyle H$

Can we show that $\displaystyle \eta(g)=g$ for all $\displaystyle g\in G$ so that we can identify $\displaystyle G$ as a subgroup of $\displaystyle H$ and $\displaystyle \eta$ is then just the inclusion?