You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$.

In fact there is a canonical isomorphism of stacks $\overline M_{0,4}(B\mathbf Z/2) \cong X(2)$, where $X(2)$ is the modular curve for the full level 2 congruence subgroup, parametrizing elliptic curves with a basis of their 2-torsion. This is easy to show on the interior. To get the result also on the boundary, use the modular interpretation of the cusps of $X(2)$ described in Deligne--Rapoport. For any $T \to X(2)$, you get a family $E \to T$ of generalized elliptic curves, and inversion in the group structure defines an admissible double cover $E \to E/\langle \pm 1\rangle \cong \mathbf P^1_T$ branched over four points, hence a $T$-point of $\overline M_{0,4}(B\mathbf Z/2)$.

You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$.

In fact there is a canonical isomorphism of stacks $\overline M_{0,4}(B\mathbf Z/2) \cong X(2)$, where $X(2)$ is the modular curve for the full level 2 congruence subgroup, parametrizing elliptic curves with a basis of their 2-torsion. This is easy to show on the interior. To get the result also on the boundary, use the modular interpretation of the cusps of $X(2)$ described in Deligne--Rapoport. For any $T \to X(2)$, you get a family $E \to T$ of generalized elliptic curves, and inversion in the group structure defines an admissible double cover $E \to E/\langle \pm 1\rangle$ branched over four points where $E/\langle \pm 1\rangle$ is a rational curve, hence a $T$-point of $\overline M_{0,4}(B\mathbf Z/2)$.

You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$.

In fact there should be a canonical isomorphism of stacks $\overline M_{0,4}(B\mathbf Z/2) \cong X(2)$, where $X(2)$ is the modular curve for the full level 2 congruence subgroup, parametrizing elliptic curves with a basis of their 2-torsion. This is easy to show on the interior and I think it will be true along the boundary too, using the modular interpretation of the cusps of $X(2)$ described in Deligne--Rapoport.

You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$.

In fact there is a canonical isomorphism of stacks $\overline M_{0,4}(B\mathbf Z/2) \cong X(2)$, where $X(2)$ is the modular curve for the full level 2 congruence subgroup, parametrizing elliptic curves with a basis of their 2-torsion. This is easy to show on the interior. To get the result also on the boundary, use the modular interpretation of the cusps of $X(2)$ described in Deligne--Rapoport. For any $T \to X(2)$, you get a family $E \to T$ of generalized elliptic curves, and inversion in the group structure defines an admissible double cover $E \to E/\langle \pm 1\rangle \cong \mathbf P^1_T$ branched over four points, hence a $T$-point of $\overline M_{0,4}(B\mathbf Z/2)$.