The fixed point property of a poset and the fixed point property of the poset induced by its extremal points

For a connected finite poset $P$, let $E(P)$ be the poset induced by the extremal points of $P$. We show that the fixed point property of $E(P)$ implies the fixed point property of $P$. On the other hand, we show that a homomorphism $f : E(P) \rightarrow Q$ can be extended to $P$ if $Q$ is a flat poset not containing a 4-crown. We conclude that every retract-crown of $E(P)$ with more than four points is a retract-crown of $P$, too. We see that for $P$ having the fixed point property but $E(P)$ not, every edge of every crown in $E(P)$ must belong to a so-called improper 4-crown, with additional specifications if $P$ has height two. The results provide several sufficient and necessary conditions for $P$ having the fixed point property, and these conditions refer to objects simpler than $P$.