Given a simplicial commutative semigroup:

(1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group?

(2) is the constant simplicial set on a set, Kan fibrant?

A positive answer to (2) would give a negative answer to (1), since the constant simplicial semigroup that is degreewise the natural numbers would be Kan fibrant and not a simplicial group.