Malcev endomorphism rigidity theorem for 1-connected nilpotent real Lie group?

Question

This question might be really stupid, but I couldn't figure out the answer.

Let $G$ be a 1-connected real nilpotent Lie group, and $\Gamma$ be a lattice in $G$. By Malcev, every automorphism $f$ on $\Gamma$ extends uniquely to an automorphisms $\overline{f}$ on $G$. (This is a special case of Malcev's automorphisms rigidity theorem).

What if $f$ is endomorphism instead?

1. Does it extend to an endomorphism $\overline{f}$ on $G$? Why?
2. Is the extension unique? Why? (In other words, is there an endomorphism rigidity theorem on $G$?)

Thanks!