Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence".

Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be 1 if $-I\in\Gamma$, and 0 otherwise. Let $d$ be its index of $\Gamma$ inside $SL(2,\mathbb{Z})$, let $n$ be its number of cusps, and let $w_1,\ldots,w_n$ be the width of its cusps. By the signature of $\Gamma$, I mean the tuple $(c_2,c_3,c_{-1},d,n,\{w_1,\ldots,w_n\})$.

Now suppose $\Gamma_1,\Gamma_2$ are two such finite index subgroups. Let $H$ be the upper half plane. I'd like to understand the relationship between the statements:

(A) $\Gamma_1$ is conjugate to $\Gamma_2$ in $SL_2(\mathbb{Z}$).

(A') The images of $\Gamma_1,\Gamma_2$ in $PSL_2(\mathbb{Z})$ are conjugate.

(B) $\Gamma_1$ and $\Gamma_2$ have the same signature.

(C) $H/\Gamma_1$ is isomorphic to $H/\Gamma_2$ as Riemann surfaces.

(C') $H/\Gamma_1$ is isomorphic to $H/\Gamma_2$ as complex orbifolds.

(D) $H/\Gamma_1$ is isomorphic to $H/\Gamma_2$ as Riemann surfaces over $H/SL(2,\mathbb{Z})$.

(D') $[H/\Gamma_1]$ is isomorphic to $[H/\Gamma_2]$ as complex orbifolds over the orbifold quotient $[H/SL(2,\mathbb{Z})]$.

Certainly (A) implies everything else, and is equivalent to (D').

Certainly (D') $\Rightarrow$ (D), (C') $\Rightarrow$ (C), (D) $\Rightarrow$ (C) and (D') $\Rightarrow$ (C').

Also I've computed some explicit examples showing that (B) $\not\Rightarrow$ (A).

There are some other obvious implications, but mostly other than this I know very little. In particular, what does knowing the signature really tell us? What is the relationship between (B) and any of the other properties?

Does assuming (B) change any of the relationships? For example, certainly neither (C) nor (C') imply (D) or (D'), but does (B)+(C) or (B)+(C') imply (D) or (D')?