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 SωSω in a topological group GG admitting a functorial embedding [0,1]⊂G[0,1]⊂G. The latter means that each autohomeomorphism of [0,1][0,1] extends to a continuous homomorphism of GG. 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 XX containing a topological copy of the space QQ of rationals give topological groups containing SωSω.