The congruence subgroup $\Gamma_{0}(n) \subset PSL_{2}(\mathbb{Z})$ is normalized by the Fricke involution $F_n: z \mapsto -1/nz$ and so we may form the Fricke modular group $\Gamma_{0}^{+}(n)\langle \Gamma_{0}(n), F_n \rangle \subset PSL_{2}(\mathbb{R})$. (Sometimes people focus on the case $p=n$, since then I think that $\Gamma_{0}^{+}(p)$ is the full normalizer of $\Gamma_{0}(p) \subset PSL_{2}(\mathbb{R})$.)

Questions:

Is it known that and how $\Gamma_{0}^{+}(n)$ splits as a free product of cyclic groups? (This is known for $\Gamma_{0}(n)$ and I can check it by hand for $n=2$.)

Are convenient fundamental domains for $\Gamma_{0}^{+}(n)$ acting on the upper half-plane $\mathbb{H}$ known?

Are the orbifold points of $\mathbb{H}/\Gamma_{0}^{+}(n)$ completely understood?

For questions of the form `Is it known or understood... ?', I'm looking for answers of the form 'Yes, this is well-known and works as follows... or is written down here...' or 'Yes, it is well-known, but scattered over the literature...' or 'No, this is not obvious.' Also, any good reference is welcome, since I've only found bits and pieces addressing these questions.

somefundamental domain by writing down cosets for $\Gamma_{0}(n)$, but it would be nice to have a fundamental domain described by (say) inequalities. Also, is it clear what to do once you add in the Fricke involution? I apologize if this obvious, but I am new at this business. $\endgroup$1more comment