Fréchet–Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet–Urysohn fan Sω in a topological group G admitting a functorial embedding [0,1]subset ofG. The latter means that each autohomeomorphism of [0,1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space image of rationals give topological groups containing Sω.