The structure of the normalisers of the congruence subgroups of the Hecke group G 5

Let λ = 2cos (π/5) and let τ ∈ ℤ[λ]. Denote the normaliser of G 0 (τ) of the Hecke group G 5 in PSL 2 (ℝ) by N ( G 0 (τ)). Then N ( G 0 (τ)) = G 0 (τ/ h ), where h is the largest divisor of 4 such that h 2 divides τ. Further, N ( G 0 (τ))/ G 0 (τ) is either 1 (if h = 1), ℤ 2 × ℤ 2 (if h = 2) or ℤ 4 × ℤ 4 (if h = 4).